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## Material Information- Title:
- Local assessment of perturbations
- Creator:
- Hartless, Glen Lawson, 1971-
- Publication Date:
- 2000
- Language:
- English
- Physical Description:
- viii, 202 leaves : ill. ; 29 cm.
## Subjects- Subjects / Keywords:
- Acceleration ( jstor )
Curvature ( jstor ) Inference ( jstor ) Linear regression ( jstor ) Matrices ( jstor ) Modeling ( jstor ) Statistical discrepancies ( jstor ) Statistical estimation ( jstor ) Statistical models ( jstor ) Statistics ( jstor ) Dissertations, Academic -- Statistics -- UF ( lcsh ) Statistics thesis, Ph. D ( lcsh ) - Genre:
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theses ( marcgt ) non-fiction ( marcgt )
## Notes- Thesis:
- Thesis (Ph. D.)--University of Florida, 2000.
- Bibliography:
- Includes bibliographical references (leaves 194-201).
- General Note:
- Printout.
- General Note:
- Vita.
- Statement of Responsibility:
- by Glen Lawson Hartless.
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LOCAL ASSESSMENT OF PERTURBATIONS By GLEN LAWSON HARTLESS A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2000 Copyright 2000 by Glen Lawson Hartless To Christy ACKNOWLEDGMENTS I would first like to thank my chairman, Jim Booth, and cochairman, Ramon Littell, for all of their help and guidance. Their combined knowledge, insight, and mentoring has been extraordinary. I would also like to express my gratitude to them for allowing me to pursue avenues of research that are probably closer to my heart than theirs. Thanks also go to the other esteemed professors who served on my supervisory committee: James Algina, Randy Carter, and James Hobert. I would also like to acknowledge people who have influenced me academically and professionally over the years: Chuck Nowalk, John Grant, and many others in the Phillipsburg and Lopatcong school systems, Walter Pirie and the faculty at Virginia Tech, Ron DiCola at AT&T/Lucent, Larry Leemis at the College of William & Mary, Danny Perkins, Bo Beaulieu, and many others at the University of Florida, and also fellow students Jonathan Hartzel, Phil McGoff, and Galin Jones, among many others. I would also like to thank Allan West for maintaining an extremely reliable computing environment. Thanks also go to my parents for their love and support and for instilling in me the discipline needed to achieve my goals. Finally, my utmost thanks go to my wife, Christine. I am forever grateful for her unequivocal love, unending patience, and thoughtful insight. TABLE OF CONTENTS page ACKNOWLEDGMENTS ................. ABSTRACT .... ..... .... ..... .... .. .. .. .. . CHAPTERS 1 INTRODUCTION TO LOCAL INFLUENCE ............ Prem ise . . . . . . . . . . Measuring the Effects of Perturbations . Local Influence Diagnostics ......... Perturbations in Regression ......... Basic Perturbations ............. Brief Literature Review ........... O utline . . . . . . . . . . 2 COMPUTATIONAL ISSUES IN LOCAL INFLUENCE ANALYSIS 21 2.1 Curvatures and the Direction of Maximum Curvature .... 21 2.2 Local Influence Analysis of a Profile Likelihood Displacement 23 2.3 Building Blocks of Local Influence for Regression ....... 24 2.4 Perturbations to the Response in Regression ........ 27 2.5 Unequal Measurement Precision Applied to Regression . 31 2.6 Benchmarks for Perturbation Analyses . . . . .... ..36 2.7 Local Assessment Under Reparameterizations . . . ... ..40 3 PERTURBATIONS TO THE X MATRIX IN REGRESSION . 49 3.1 Perturbations to Explanatory Variables . . . . ... ..49 3.2 Perturbations to a Single Column . . . . . . .... .. 50 3.3 Perturbations to a Single Row . . . . . . . .... .. 59 3.4 Perturbations to the Entire Design Matrix . . . .... ..65 4 LOCAL ASSESSMENT FOR OPTIMAL ESTIMATING FUNCTIONS 87 Inferential Platform . . . . . . . . . . Local Assessment of Perturbations . . . . . . Acceleration, Curvature and Normal Curvature . . . Algebraic Expressions for Local Assessment . . . . Computational Formulas for J and P . . . . . . .. 88 . . 93 . .. 96 . . 98 . .. .103 . . . iv 3 8 3 8 . . . 18 . . . 18 . . . 20 4.6 Local Assessment for Maximum Likelihood Inference . . 110 4.7 First-order Assessment of the Log Generalized Variance . 116 4.8 Local Assessment for Quasi-likelihood Inference ....... .125 5 LOCAL ASSESSMENT OF PREDICTIONS . . . . . .... ..134 5.1 Prediction via First Principles . . . . . . . ... .. 134 5.2 Predictive Likelihood . . . . . . . . . ... .. 136 5.3 Influence Measures for Prediction . . . . . . ... .. 138 5.4 Estimation and Prediction in Linear Mixed Models . . . 140 5.5 Local Influence Methods for Linear Mixed Models . . . 146 5.6 One-way Random Effects Model . . . . . . ... .. 151 5.7 Residual Variance Perturbation Scheme . . . . .... ..154 5.8 Cluster Variance Perturbation Schemes . . . . .... ..159 5.9 Application to Corn Crop Data . . . . . . ... .. 167 6 MULTI-STAGE ESTIMATIONS AND FUTURE RESEARCH . 185 6.1 Local Assessment in Multi-stage Estimations . . . ... ..185 6.2 Synopsis and Additional Future Research . . . . .... ..186 APPENDICES A MODEL FITTING INFORMATION FOR SCHOOL DATA . . 188 B MODEL FITTING INFORMATION FOR CORN CROP DATA . 191 REFERENCES . . . . . . . . . . . . . . . . ... .. 194 BIOGRAPHICAL SKETCH . . . . . . . . . . . . ... .. 202 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy LOCAL ASSESSMENT OF PERTURBATIONS By Glen Lawson Hartless August 2000 Chairman: James G. Booth Major Department: Statistics Statistical models are useful and convenient tools for describing data. However, models can be adversely effected by mistaken information, poor assumptions, and small subsets of influential data. Seemingly inconsequential changes to the observed data, the postulated model, or the inference technique may greatly affect the scientific conclusions made from model-fitting. This dissertation considers local assessment of perturbations in parametric and semi-parametric models. The effects of perturbation are assessed using a Taylor-series approximation in the vicinity of the fitted model, leading to graphical diagnostics that allow the analyst to identify sensitive parameter estimates and sources of undue influence. First, computational formulas are given to assess the influence of perturbations on linear combinations of the parameter estimates for maximum likelihood. A diagnostic plot that identifies sensitive fitted values and self-predicting observations in linear regression is introduced. Second, the eigensystem of the acceleration matrix for perturbing each element of the design matrix in linear regression is derived. This eigensystem leads to a diagnostic plot for identifying specific elements that are influential, and it is shown how the results relate to principal components, collinearity, and influence diagnostics. Next, it is shown that the conceptual framework and computational tools easily extend from maximum likelihood to inference that is performed using estimating equations. In particular, local assessment of parameter estimates and standard errors derived from optimal estimating functions is addressed. Algebraic expressions and computational formulas are given for both first- and second-order assessment, and applications such as quasi-likelihood are considered. Finally, the techniques are applied to linear mixed models in order to assess the influence of individual observations and clusters. While previous applications considered only estimation, the effects of perturbations on both estimation and prediction is addressed. It is shown that the effects on the fixed and random effect inferences can be assessed simultaneously by using an influence measure based on the unstandardized predictive log-likelihood of the random effects. CHAPTER 1 INTRODUCTION TO LOCAL INFLUENCE 1.1 Premise George Box (1982, p. 34) stated "all models are wrong; some models are useful," and A. F. Desmond (1997b, p. 117) stated "a robust method ought not to perform poorly in neighborhoods of the nominally true model." Indeed, conclusions based on a statistical model should be reliable under small perturbations to the data, the model's assumptions or the particulars of the estimation method. In a seminal paper, Cook (1986) utilized the basic concept of perturbing the postulated model or the observed data to develop a variety of diagnostic tools. He showed how small perturbations can be used to develop easily-computable diagnostics for almost any model estimated by maximum likelihood (ML) estimation. 1.2 Measuring the Effects of Perturbations In a statistical analysis, parameter estimates are a crucial part of the decision-making process. These estimates are a function of the observed data, the postulated model, and the analyst's choice of statistical inference technique. Any one of these three aspects of a statistical-modeling situation is subject to uncertainty and doubt: observed data may be incorrectly recorded or subject to rounding error, a slightly different model may be just as plausible as the postulated model, and nuisance aspects of modeling such as choosing hyperparameters are subject to second-guessing. If small changes in these aspects of a statistical modeling problem create large changes in parameter estimates, the analysis is unreliable. Therefore, the change in parameter estimates due to perturbation can serve as a barometer for the robustness of the scientific conclusions. Following Cook (1986), a vector of q perturbations is denoted as w = (WO1, W2,... wq)', and the q-dimensional space of possible perturbations is denoted as 0. Included in this space is one point, wo, that does not change the original formulation of the problem, and this is called the null perturbation. The perturbations are not parameters to be estimated, but rather they are known real-valued quantities introduced into the problem by the analyst. These perturbations might represent measurement errors, unequal variances, case weights, correlated errors, missing predictors or some other change to the data, model or inference method. Prudent choice of perturbations can lead to useful diagnostics. For example, if parameter estimates are sensitive to perturbing the contribution (weight) of a particular observation to estimation, it indicates that the observation is overly influential on the estimates. For data y and parameters 0, let L(O; y) denote the log-likelihood from the original postulated model, and let L,(0; y) denote the log-likelihood resulting from the inclusion of perturbations. If the perturbation specified is the null perturbation, then the two likelihood are equivalent. The resulting maximum likelihood estimators (MLEs) of these two likelihood are denoted as 0 and 06, respectively. Cook (1986) used the likelihood displacement to summarize parameter estimate changes under perturbation: LD(w) = 2[L(0; y)- L(06; y)]. (1.1) From inspection of (1.1), we see that the likelihood displacement is the difference of the original likelihood evaluated at two values: the original MLEs and the perturbed MLEs. Because 0 maximizes the original likelihood, LD(w) is non-negative and achieves its minimum value of zero when w = w0. As the perturbed MLEs become increasingly different from the original MLEs, LD(w) becomes larger. Thus, the size of LD(w) provides a measure of how much the perturbation affects parameter estimation. Effects on a subset of parameters, 01 from 0' = (0', 0'), can be obtained by using the profile likelihood displacement: LD[I"w) = 2[L(01, 02; y) L(01,, g(06y); y)], (1.2) where 0 is from = (0 2.), and g(0}1) is the vector that maximizes the profile log-likelihood, L(01,, 02), with respect to 02. The profile likelihood displacement allows influence to be "directed" through 01, with the other parameter estimates only changing in sympathy. In the next section it is shown how to obtain diagnostic tools from a local influence analysis of the likelihood displacement and the profile likelihood displacement. 1.3 Local Influence Diagnostics Although we could simply choose values for w and compute LD(w), this is not particularly useful for two reasons. First, LD(w) is often not a closed-form function of w. Hence, if we wish to find its value, re-fitting the model may be necessary. Second, even if calculating LD(w) was easy, we must somehow decide what values to use for w. This choice would be highly data-dependent and a nuisance for the analyst. Therefore, we would like a methodology that can examine the information contained in the function LD(w) and yet does not require a choice of actual values for w nor involve re-estimating the model parameters. Following Cook (1986), this can be accomplished using the notion of curvature. Simple plots can then be used to identify whether estimation is sensitive to any particular elements of the q-dimensional perturbation. 1-f ----- Figure 1.1: Schematic of a plane curve with its tangent line. The angle between the tangent line and the x-axis is . 1.3.1 Curvature of a Plane Curve Curvature is a measure of how quickly a curve turns or twists at a particular point (Swokowski 1988). Let (f(t), g(t)) be a smooth plane curve. The arc length parameter s of such a curve is the distance travelled along the curve from a fixed point A to another point on the curve (Figure 1.1). Letting tA denote the value of t that yields point A, s can be computed as s(t) = (f(u))2 + (gt(u))2)]du, (1.3) and we may express the curve in terms of s: (f(s-l(s)), g(s-l(s))). Additionally, let T(s) be the tangent line at a point on the curve, and let q denote the angle between T(s) and the x-axis. Curvature is defined as the absolute value of the rate of change in q with respect to s: C 0 (1.4) A plane curve that is turning quickly at (f(t), g(t)) has a larger curvature than a curve that is nearly straight at that point. 2 1.5 0.5 12 -0.5 -1 -1.5 -2 Figure 1.2: Curve with two circles of curvature The reciprocal of curvature is called the radius of curvature, and it is the radius of the best fitting circle at that point on the curve. These circles of curvature have the same first and second derivatives and tangent line as the curve and lie on its concave side. Figure 1.2 shows the curve (j sin(t),1 sin(2t)) and two of its circles of curvature. The one on the left occurs when t = 3 while the other one occurs 2 when t = arccos(O). Their curvatures are 6 and 8/9 respectively. The expression "turning on a dime" refers to to a sharp turn: one in which the radius of curvature is small, and consequently the curvature of the path is large. The formula for curvature is f'(g"(t) g'(t)f"(t) C =---------3-.(1.5) ((f'(t))2 + (g(t))2)3 We are particularly interested in assessing curvature at the local minimum of a function from x to y, i.e. when x = t. For this special case, curvature measures how quickly the function increases from that minimum value. In this case, (1.5) simplifies to the acceleration of the curve: S19X2g) n( 0 = 2g,~ = (1.6) C x x=xo Figure 1.3: Likelihood displacement with directional vectors where g(xo) is the function's minimum. This result can be applied to LD(w) at w0 when there is only a single perturbation, w. Here, the curvature is given by O2LD(w) (1.7) Ow2 Ww=o(1 and it measures how quickly the parameter estimates change when w is moved away from the null perturbation. A large curvature indicates that even if w is close to, or local to w0, the likelihood displacement increases substantially. Hence, a large curvature indicates that even a slight perturbation can be very influential on parameter estimates. 1.3.2 Using Plane Curves to Describe LD(w) The curvature of plane curves can also be used to describe LD(w) when q > 1. Now w can be moved away from wo in different directions, as shown by the vectors in Figure 1.3 for a two-dimensional perturbation. These vectors can be written as LO,(a) = wo+ av, (1.8) Figure 1.4: Intersection of plane and surface creates space curve where the vector v determines the direction and the value a determines the position relative to wo. Let P, denote a plane that contains w,(a) and extends directly upward from 0 to intersect LD(w) (Figure 1.4). The intersection of P" and LD(w) forms a space curve, (w,(a)', LD(w,(a)))1 Because this space curve is contained within the plane P,, its curvature can be obtained as if it were a plane curve, (a, LD(aw,(a)). Thus the curvature of LD(w,(a)) at w0 is simply Cv 2LD( (a)) (1.9) Oa2 La=O Different directions produce curvatures of various sizes. The normalized vector that produces the largest curvature is denoted Vmax and is of particular interest. 1.3.3 Direction of Maximum Curvature, Vmax The relative sizes of the elements of vmax indicate which elements of w are most influential on the parameter estimates. For example, if the ith element of Vmax 1 More formally, these space curves are called normal sections. The term comes from the fact that each plane, Ps, contains the surface's principal normal vector, the vector that is orthogonal to the surface's tangent plane at Wo. Curvatures of the normal sections are referred to as normal curvatures. is large, then wi is a key player in producing the largest curvature of LD(w) at the null perturbation, i.e. in producing an immediate change in parameter estimates. The direction of maximum curvature is the main diagnostic of Cook's local influence methodology. Typically, Vma is standardized to have length one and is then plotted against indices 1, 2,..., q. If any elements are particularly large relative to the others, they stand out in the plot. The analyst then may want to take steps to alleviate the sensitivity to the perturbation. What, if any, steps to be taken depend upon the perturbation scheme and the particulars of the data analysis. For some models and perturbation schemes, the formula for Vmax is known explicitly, while in other situations it requires some moderate computations. In analyses such as regression, Vmax is typically a function of familiar quantities. 1.4 Perturbations in Regression Consider the normal linear regression model: Y = XO + C, (1.10) where Y is a n x 1 vector of responses, X is a known n x k model matrix of rank k, ,3 is a k x 1 vector of unknown parameters, and e is a n x 1 vector of independent identically distributed normal random variables with mean 0 and variance or2 > 0. Hence, Y N(X ,oa2In), and 0 = (01,...,3k, a2)'. 1.4.1 Scheme 1: Measurement Error in the Response In order to examine the effects of measurement error in the response, Schwarzmann (1991) introduced perturbations to the observed data. He considered fitting a model to the observed data set with slight perturbations: y, + wi, for i = 1,..., q. Hence, w0 is a vector of zeroes for this perturbation scheme. By letting q = n, all of the observations can be perturbed simultaneously. This allows the Table 1.1: Scottish hills race data Observation Record Ti 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 Source: Atkinson (1986) me (min.) 16.083 48.350 33.650 45.600 62.267 73.217 204.617 36.367 29.750 39.750 192.667 43.050 65.000 44.133 26.933 72.250 98.417 78.650 17.417 32.567 15.950 27.900 47.633 17.933 18.683 26.217 34.433 28.567 50.500 20.950 85.583 32.383 170.250 28.100 159.833 Distance (miles) 2.5 6.0 6.0 7.5 8.0 8.0 16.0 6.0 5.0 6.0 28.0 5.0 9.5 6.0 4.5 10.0 14.0 3.0 4.5 5.5 3.0 3.5 6.0 2.0 3.0 4.0 6.0 5.0 6.5 5.0 10.0 6.0 18.0 4.5 20.0 Climb (ft) 650 2500 900 800 3070 2866 7500 800 800 650 2100 2000 2200 500 1500 3000 2200 350 1000 600 300 1500 2200 900 600 2000 800 950 1750 500 4400 600 5200 850 5000 analyst to determine which observations would have the most effect on parameter estimates in the event that there is measurement or rounding error. 1.4.1.1 Influence on the full vector of parameter estimates Atkinson (1986) analyzed a data set of record times for 35 different hill races, as recorded by the Scottish Hill Runners Association (Table 1.1). Scatter plots indicate the positive association of the three variables: winning time, race distance and race climb. (Figure 1.5). The influence of perturbations on the parameter estimates from a regression of winning time on DISTANCE and CLIMB is of interest. For convenience, the regressors and response have been centered and scaled, thereby eliminating the need for an intercept. Thus, the full vector of p parameters is 0 = (ldist,I Acimb, a2)'. Figure 1.6 shows vmax plotted against the indices 1 through q = n = 35 for the Scottish hills data. Element 18, which corresponds to perturbation of the 18th observation, is the dominant contributor to the direction of maximum curvature. This observation is from a short race with little CLIMB, and yet it has a large record time. Similar races have much smaller record times, indicating that race 18 is an outlier. Atkinson (1986) has actually suggested that the winning time for this race may have been 18.67 minutes rather than 1 hour and 18.67 minutes. Interestingly, it can be shown that the direction of maximum curvature for Scheme 1 is proportional to the residuals. Thus, if we standardized the vector of residuals to length 1 and plotted them against case indices, we would obtain Figure 1.6. This implies that parameter estimates are most sensitive to a set of small additive measurement errors when those errors tend to exaggerate (or correct) the model's error in estimating each data point, with the most exaggeration (or correction) occurring for the most poorly predicted points. .18 1, 043 5 10 15 20 25 30 D~ac (a) Winning distance time vs. race 01) 50 e. 1000 2000 3000 4000 5000 6000 7000 80061imb (b) Winning time vs. race climb Climb 8000 a7 7000 6000 5000 4000 3000 2000 "' *11 *0 1000 J ell 1 5 10 15 20 25 30 Distance 5 10 15 20 25 30 (c) Race climb vs. race dis- tance Figure 1.5: Hills data; scatter-plots with highlighted cases 1 0.75 84 0.5 4 0.25 o 0 0 -0.25 -0.5 -0.75 0 5 10 15 20 Case 25 30 35 Figure 1.6: Hills data; response perturbation; Vma. for 0 0 0 a ** ; O0 - 12 I 1 0.75 0.75 0.5 0.5 0.25 a 0.25 * ,.' .. .' .. .. .. '' --- .*,0>* '* "al -- * i* n. ... *5 'a -0.25 -0.25 -0.5 -0.5 -0.75 -0.75 1I -1 (a) Vmax for directed influ- (b) vmax for directed influ- ence on /dist ence on climb Figure 1.7: Hills data; response perturbation 1.4.1.2 Influence on a subset of parameter estimates Plotting Vmax for the profile likelihood displacement, LD011 (w), can indicate which components of w affect a subset of parameters. Examining influence on 91 = a2 results in the same vm,,a as the original likelihood displacement, i.e. it is proportional to the residuals. The directions of maximum curvatures for 01 = /dist and 01 = dcimb are shown in Figures 1.7(a) and 1.7(b). The first plot indicates that small measurement error in race 11's winning time is most influential on A&ist. In the plot of v.ax for fliimb, elements 7 and 11 stand out. The fact that the two elements have opposite signs can be used for interpretation. They indicate that maximum curvature is primarily achieved by perturbing winning times 7 and 11 in opposite directions (i.e. making one observation smaller and the other larger). Thus, if the record winning times for races 7 and 11 were slightly closer together or farther apart, this would affect the slope estimate for CLIMB. 1.4.1.3 Plotting options Note that the standardized Vma,,, is only unique up to sign change. For example, Figure 1.8 displays the same standardized vector as Figure 1.6, but with all of the elements' signs switched. This suggests an alternative plot in which the 1 0.75 0.5 0.25 o o 025 1" 15 ~ 2r O 30 *35 -0.25 O'I -0.5 -0.75 -1 Figure 1.8: Hills data; alternate rendering of vma.,, signs of the elements are suppressed: a star plot. Here, each element is displayed as a point, with the distance from the origin equal to its absolute value. The q elements are simply displayed around the point (0,0), starting in the first quadrant and ending in the fourth quadrant. The plot is completed by labelling the points with indices. Labeling all points can result in cluttered plot (1.9(a)), and so a better option is to only label the largest elements (1.9(b)). In these renderings, element 18 is again noticeably large, but our attention is also drawn to element 7. This observation has the characteristics of an influential observation: it has the highest climb and one of the longest distances (Figure 1.5(c)), as well as the second largest residual. Unlike the index plot, the star plot does not give the illusion that the indices are in a specific order. However, the suppression of element signs can be an advantage or a disadvantage. The advantage is that when looking at several plots of directional vectors, suppressing the signs helps the analyst identify patterns by emphasizing element size. The disadvantage is that the diagnostic interpretation of the signs is lost. 1- 1- 0.5- 0.5- 7 7 5 2 35 0.5 1 1 0.51 18 13 18 26 31 -0.5- -0.5- -1. -1- (a) All elements numbered (b) Largest elements numbered Figure 1.9: Hills data; IVmaxl as a star plot 1.4.2 Scheme 2: Perturbations to Measurement Precision Another assumption that can be perturbed is that of equal variance (Cook 1986). This is accomplished by letting V(yi) = for i = 1,... n. Here the null perturbation is w0 = In. With this perturbation scheme, we deem that some observations are observed with more precision than others. The resulting perturbed MLEs weight each case according to its perturbed variance. If certain cases are influential, then even a slight down-weighting or up-weighting of them produces large displacements. This perturbation scheme has an important connection to the widely-used influence measure Cook's Distance (Cook 1977). Recall that the ih Cook's Distance is a measure of the change in regression parameter estimates that results from deleting the ith observation. Clearly a deleted observation provides no information for finding 3. Similarly, as the ith perturbation is decreased from 1, the ith observation provides less and less information to the estimation process owing to its increased variance. Thus, case-deletion can be thought of as a rather drastic perturbation: one that results in a variance -*0 oo for the ilh observation. Case-deletion is referred to as a global influence technique because it assesses the effect of perturbations that are not local to the null perturbation in fl. 1.4.2.1 Influence on the full vector of parameter estimates Figure 1.10(a) shows the direction of maximum curvature for the Scottish hills data set. Again, element 18 dominates the direction of maximum curvature. This implies that the precision of observation 18 is the most influential on the full set of parameter estimates. 1.4.2.2 Influence on &2 Using the profile likelihood displacement, influence directed upon a2 alone can be assessed. Observation 18 is again implicated by vmax as being most locally influential (Figure 1.10(b)). 1.4.2.3 Influence on individual regression parameter estimates Figure 1.10(d) shows Vma,, when isolating effects on lMist. Although there are several points that stand out, observation 7 is identified as being most influential since element 7 is large. This race is also implicated as being locally influential in the plot of Vmax for /dimb (Figure 1.10(c)). In fact, the DFBETAS diagnostic (Belsley et al. 1980) also identifies race 7 as being influential on cdiimb. Recall that the value of DFBETAS$j measures how the jth parameter estimate is affected by deleting the ith observation. Writing the n DFBETAS for the jth parameter estimate in vector form, it is shown in Section 2.5 that this vector is proportional to the following Hadamard (element-wise) product of vectors: .. - ...-. . . . - 0.25 10 35 -0.25 -0.5 -0.75 -1 ..5 10 6o 15 2 0 25 30 35 (a) Vmax for 6 I !~i~ 2T'T~30 ~ S (c) Vmax for climbb (b) Vmax for &2 5 Ct O e0 25 30 35 p (d) Vmax for fdist 1 0.75 0.5 0.25 S -0.25 -0.5 -0.75 -1 0 9 '.' *,:* S0'''a'5"'3'0 935 Op* (e) Vmax for / (f) v2 for 4 Figure 1.10: Hills data; variance perturbation I 5 10 w 15 TOT 75 1 0.75 0.5 0.25 -0.25 -0.5 -0.75 *i Figure 1.11: Standardized vector of DFBETAS for cdimb 1 1hn rl rlj 1-hil 1 1-h2 r2 r2J (1.11) 1 1- ) f r. rj Here, hi is the ith diagonal element of the "hat" matrix X(X'X)-1X', ri is the ih residual, and rij is the ill residual from regressing Xj on the other columns of X. It can be shown (see Section 2.5) that vma,, for directing the effects of the variance perturbation on 4j is proportional to ri rij r2 r2j (1.12) r. v L 2 r, rni The elements of the two vectors differ only by a factor of 1 In fact, standardizing the vector of 35 DFBETAS for 4c3imb to length 1 and plotting them (Figure 1.11) produces a figure nearly identical to Figure 1.10(c). Influence on both regression parameter estimates Figure 1.10(e) displays the direction of maximum curvature for directed influence on 6. Plot (f) is the direction of largest curvature under the constraint of orthogonality to Vma,. The two plots are very similar to the directed influence plots in (c) and (d). 1.5 Basic Perturbations As an alternative to finding the direction of maximum curvature, we may utilize basic perturbations. The i1th basic perturbation is the same as the null perturbation except for an additive change of 1 in its ith element. For example, the first basic perturbation is w0 + (1,0,..., 0)', and corresponds to moving a distance of 1 along the first axis in fl. Typically, the curvatures for each of the q basic perturbations would be computed and then plotted together. As an example, consider applying the variance perturbation to a linear regression, and examining directed influence on J3. In this case, the ith basic curvature, Cb,, is a local influence analog to the ith Cook's Distance, because it corresponds to re-weighting a single observation. It is shown in Section 2.7.2.1 that the ith basic perturbation for this scheme measures influence on the ith fitted value. Figure 1.12(a) plots the 35 curvatures for the basic perturbations applied to the Scottish hills data, while Figure 1.12(b) plots the 35 Cook's Distances. Although the scales differ, they provide similar information. Both plots draw attention to race 7, but they differ somewhat in their emphasis of races 11 and 18. 1.6 Brief Literature Review Local influence techniques have been applied to a number of statistical models, most notably univariate and multivariate linear models (Cook 1986; Cook and Weisberg 1991; Lawrance 1991; Schwarzmann 1991; Tsai and Wu 1992; Wu and Luo 1993a; Kim 1995; Kim 1996c; Tang and Fung 1996; Fung and Tang 1997; 2.5 . 3 2 1.5 2 0.5 1 3 .p.*. *.. l...i. O,. . ...pg..s... 0... . 0 15 2O 25 30 35 5 10 15" 02"o 5- 30 3 (a) Basic curvatures (b) Cook's Distances Figure 1.12: Hills data; Cb's for variance perturbation and Cook's Distances Kim 1998; Rancel and Sierra 1999). A number of articles have also appeared on applications in generalized linear models (GLMs) (Cook 1986; Thomas and Cook 1989; Thomas and Cook 1990; Thomas 1990; Lee and Zhao 1997) and survival analysis (Pettitt and Daud 1989; Weissfeld and Schneider 1990; Weissfeld 1990; Escobar and Meeker 1992; Barlow 1997; Brown 1997). Other applications include linear mixed models (Beckman et al. 1987; Lesaffre and Verbeke 1998; Lesaffre et al. 1999), principal components analysis (Shi 1997), factor analysis (Jung et al. 1997), discriminant analysis (Wang et al. 1996), optimal experimental design (Kim 1991), structural equations (Cadigan 1995; Wang and Lee 1996), growth curve models (Pan et al. 1996; Pan et al. 1997), spline smoothing (Thomas 1991), Box-Cox transformations (Lawrance 1988; Kim 1996b), nonlinear models (St. Laurent and Cook 1993; Wu and Wan 1994), measurement error models (Zhao and Lee 1995; Lee and Zhao 1996), and elliptical linear regression models (Galea et al. 1997). Many authors have presented modifications of, extentions to, and theoretical justifications for the basic methodology (Vos 1991; Schall and Gonin 1991; Schall and Dunne 1992; Farebrother 1992; Wu and Luo 1993a; Billor and Loynes 1993; Kim 1996a; Fung and Kwan 1997; Lu et al. 1997; Cadigan and Farrell 1999; Poon and Poon 1999). Extentions to Bayesian methodology have also been considered (McCulloch 1989; Lavine 1992; Pan et al. 1996). 1.7 Outline In Chapter 2, computational formulas for local influence analysis are discussed and derived for the two perturbation schemes discussed in this chapter. Chapter 3 contains derivations and examples of local influence diagnostics for linear regression under perturbations to the X matrix. In Chapter 4, the methodology is extended to a general class of influence measures and estimation techniques. Computational tools are also provided. In Chapter 5, inference and local influence analysis for prediction problems are discussed. Techniques for assessing estimation and prediction in linear mixed models are developed and applied to data. In Chapter 6, some preliminary results that can be applied to multi-stage estimations are presented. The chapter also contains commentary on additional areas of future research. The appendices contain details for data analyses. CHAPTER 2 COMPUTATIONAL ISSUES IN LOCAL INFLUENCE ANALYSIS In this chapter, computational formulas for local influence are reviewed (Cook 1986; Beckman et al. 1987), To demonstrate the mechanics of the formulas, detailed derivations of previous results for the regression model (Cook and Weisberg 1991; Schwarzmann 1991) will be presented with minor additions. Next, the issue of how to obtain objective benchmarks is considered, and some new novel solutions are outlined. Finally, local influence is discussed under reparamaterizations of the perturbation space f and the parameter space E. By applying previous results (Escobar and Meeker 1992; Pan et al. 1997), some new diagnostic tools for influence on fitted values in linear regression are derived. 2.1 Curvatures and the Direction of Maximum Curvature Cook (1986) showed that the curvature of LD(w) in direction v can be expressed as C, = 2v'A'(-L--)Av, (2.1) where -L is the observed information matrix of the original model, -L = L(0; y) and Apxq has elements Aij = &2Lw (O; y) aOiOwj O=6,w=wo for i = 1,... ,p and j = 1,... q. This result is proved in Chapter 4 as a special case of Theorem 4.2. The q x q matrix F = 2A'(-L-)A is the acceleration matrix. It is easily obtained because A is a function of L,(0; y), and -L-1 is typically computed as part of ML estimation since it serves as an estimated asymptotic covariance matrix for the parameter estimates. Hence, computing the curvature in a particular direction is straightforward. In addition, the diagonal elements of the acceleration matrix are the q basic curvatures. Finally, it is well known how to obtain the maximum value of a quadratic form such as (2.1) (Johnson and Wichern 1992, pp. 66-68). Specifically, Cma, is the largest eigenvalue of F, and Vmax is the corresponding eigenvector. Hence, finding Cm. and Vmax is numerically feasible. The acceleration matrix can provide additional information. For example, the second largest eigenvalue is the largest curvature in a direction that is orthogonal to Vma. Hence, its corresponding eigenvector indicates a direction of large local influence that is unrelated to the first one. Additional elements of the spectral decomposition of F can be used in a similar fashion. A consequence of this derivation is that it shows how local influence techniques reduce the dimensionality of examining LD(w) via a small number of curvatures and directional vectors. It is well known that the number of non-zero eigenvalues of a symmetric matrix such as F is equal to its rank. Since it is a product of other matrices, the rank of F is no more than the minimum of their ranks, which is typically p. In the regression examples given previously, p = 3 and q = 35, meaning that the local influence approach can summarize the 35-dimensional surface LD(w) with three curvatures and directional vectors. Similarly, LD0(1(w) was summarized with just two, as shown in Figures 1.12(c) and (d). Despite the advantages of the eigen-analysis of F, one difficulty that arises is the occurence of eigenvalues with multiplicities greater than one. For example, if Cmax = A1 has multiplicity greater than 1, then there is no unique Vmax to plot. In these situations it may be more prudent to look at directed influence on single parameter estimates. 2.2 Local Influence Analysis of a Profile Likelihood Displacement The profile likelihood displacement, LD[l](w), was presented in Section 1.2 as a measure of influence directed through 01. Cook (1986) showed that the acceleration matrix can be expressed as Po"1 = -2A'(L1 B22)A, (2.2) where 0 0 B22 = 0 L-12 and L22 comes from the partition L,= ,12 L21 L22 where t3 9- 9' L -00200, o= This result is proved in Chapter 4 as a special case of Theorem 4.2. Beckman et al. (1987) considered directing influence on a single parameter estimate. They noted that in this case, the acceleration matrix is the outer product of a vector, and hence has only one non-zero eigenvalue. Without loss of generality, suppose that we wish to direct influence on the first parameter estimate in 0. The eigensystem is then given by A= 21|A'TII v oc AT, (2.3) 1 where T is the vector for which TT' = (L B22). It can be expressed as T= )1 (2.4) where c = L1 L12L21L21 2.3 Building Blocks of Local Influence for Regression In this section, the building blocks of F for a multiple linear regression are constructed. Subsequent sections apply these results to the response and variance perturbations used in Chapter 1. 2.3.1 Notation Unless otherwise noted, the derivations assume the presence of an inter- cept in the model so that the residuals sum to zero. Centering and scaling the regressors tends to faciliate interpretation, but is not necessary. The number of regressors is k = p 1. H = X(X'X)-1X. Xi represents the ith column of X, and x\ represents the ith row of X. r = (In H)y represents the residuals. r, represents the residuals from fitting a model to the ih predictor using the other predictors. These can be considered as values of an adjusted predictor (i.e. adjusted for relationships with other predictors). Note: Hr, = ri and (I H)r, = 0. X(i) represents the X matrix with the ith column removed. Similarly, 0(i) is the regression parameter vector with the ith element removed, and A(i) is A without the ith row. 2.3.2 Acceleration Matrix The ML estimates of 0 are the same as those of ordinary least squares, = (X'X)-'X'y ,and the ML estimate of &2 is r. The p x p estimated asymptotic variance-covariance matrix of the MLEs is [&2 (X'X)-1 0 01-- 2& n Partitioning A' into [A A',2], the acceleration matrix is given by F = 2A'(_-L)A = 2&2A (XX')-1A + --44A2A,2. n 2.3.3 Acceleration Matrix for Directed Influence on a2 The acceleration matrix for directed influence on 6,2 will utilize the matrix B22, which is given by 0 0 0' 2& n This implies that the acceleration matrix for directed influence on 8.2 is [2] = -2A,( 1 B22)A 46,4 (2.5) = -A-A2,. n Noting that A'2 is in fact a q x 1 vector, there will be a single eigenvalue and eigenvector given by n44 11A' V1OCAt.2 These quantities are the maximum curvature and its direction, respectively. 2.3.4 Acceleration Matrix for Directed Influence on 13 When directing influence on /3, we have B22__2 (XX')-1 0 L _B22 : Tf 0 The resulting acceleration matrix is = 2,&2A (XX')-1 Af. (2.6) Note that this implies that directed local influence on /3 is equivalent to local influence for /3 when treating a2 as known and equal to its MLE. 2.3.5 Acceleration Matrix for Directed Influence on a Single Regression Parameter By applying the general results of Beckman et al. (1987), we can construct T (2.4) for directed influence on 31 in a regression model. Using results for the inverse of a partitioned matrix, TT' = -(L B22) can be expressed as 1 *"*2 d -(l)X(1))-I X!)Xl 0 -XX(l) (X(1)X(1))1 0 (X(l)X(l))- X1)XlXlX(1)(X(1)X(1))X 0-1 , O-1 0 (2.7) where d = ,~X 1 where d = XlX1 X'X(1)(X(1)X(1)) X'()Xl. We denote (X'1)X(1)) X'1)Xl by j'1, as it is the regression parameter estimates from modeling X, as a linear combination of the other explanatory variables. This simplifies the notation of (2.7) greatly: -(L B22) 2 1Iirl12 1 -Vi0 -7i1 7i'1 0 0 0' 0 (2.8) Using these results, we find 1 T lrll (2.9) 0 and this leads to the following expressions for the maximum curvature and its direction when directing influence on 13i: Cmax = 21|A'TI| = 2 |2 A1 A/ iljl vmax oc AT oc A A^ The next two sections apply these results to document more completely the two perturbation schemes given in Chapter 1. 2.4 Perturbations to the Response in Regression The results of this section closely follow those of Schwarzmann (1991), with only minor additions. As in Chapter 1, a set of additive perturbations are introduced into the observed data to examine the influence of measurement error in the response. An n x 1 vector of perturbations is introduced such that the perturbed likelihood is based on Y + w N(X13, a2In). The null perturbation is wo0 = On. Equivalent results can be obtained by perturbing the model of the mean to be X3 w. That is, by considering a model assuming Y N(X/3 w, a2In). Hence, the measurement error perturbation can be interpreted as a type of mean-shift outlier model; each perturbation is a "stand-in" for an additional parameter that could correct or exacerbate lack-of-fit. 2.4.1 Building Blocks The perturbed likelihood is L,,(O; y) cx -- logu2 - 1(y Xf + Wy Xf3 + w) n 1 n log 02 W-(y + W)'(y + w) "i 2 -n log a - 2 1-- (' + 2w' w'w), where y = y Xf3. The next step is construction of A. Partial derivatives with respect to w are as follows: OL,(O;y)0; 1 Ow (2 +2w) 1 1 W-(y XO +W). Second partial derivatives are then 92 L,,(0; yi) OUL,(0; y) _ Oaua2 1 - x U2 1 -X03+w). Evaluating at the original model yields 902L,(O; y) OwO/3' -=6,6 =W a2L,(0; y) woao2 0=-,=wo 1 1 -r From these quantities, the n x p matrix A' can be constructed: A'= X r ] Finally, the acceleration matrix is S1 r 2 (XX')-1 0 X' F-=2- IX[2xt- 2 2H (2.10)rr' &2 n&4 8.2 2.4.2 Directed Influence on a2 The acceleration matrix for the profile likelihood displacement for &2 is simply the second term of (2.10): [2] rr', (2.11) n&.4 Solving the characteristic equations (F AI)v = 0 directly, the single eigensolution is Cmax = Amax = Vmax (c r. Therefore, as mentioned in Chapter 1, Vmax for directed influence on &2 is proportional to the residuals, 2.4.3 Directed Influence on/ Without loss of generality, consider directed influence on /13i. Rather than find the acceleration matrix, we proceed directly to determining the sole eigenvector proportional to A'T and the corresponding curvature 211A'T112, as per (2.3) and (2.9): I [ 1 1 VmaxO cx Xl X(1) i ^1 S [xl- x(l)- ] &IlrxII 1 iI,111 -< T7--'1 oc ri and C 211 1i~ r 112= 2 Cmax =2I _r__ I 2 Therefore, the direction of maximum curvature for directed influence on f3 is proportional to the jth adjusted predictor. 2.4.3.1 Directed influence on The acceleration matrix for the profile likelihood displacement for 3 is the first term of (2.10): -] H. (2.12) &~2 This matrix has a single non-zero eigenvalue of with multiplicity k, implying that there are an infinite number of directional vectors that have maximum curvature. Although not discussed by Schwarzmann (1991), one set of orthogonal solutions can be motivated by considering the canonical form of the model: Y = Za + E, (2.13) where Z=Xp W= [IW ... k I and i = the ith eigenvector of X'X. Because Wp' = Ik, this is simply a reparameterization of the original regression model. The new regressors, Z1 = X I, ..., Zk = Xpk, are known as the principal components, and they are mutually orthogonal. In addition, each satisfies the characteristic equations of F[]: 2- 2I) Z, T (H I)X i (2.14) =0. Therefore, the standardized principal component regressors can serve as an orthogonal set of directional vectors, each of which has maximum curvature. Note also that the k adjusted regressors, rl, . rk, are also directions of maximum curvature, but they do not constitute an orthogonal set. 2.4.3.2 Influence on 0 The acceleration matrix F given in (2.10) has two non-zero eigenvalues: 4 and ^, with the second one having multiplicity k. The first eigenvector is oc r, and the set of k principal components can be used to complete an orthogonal set of eigenvectors. Comfirmation of these solutions is straightforward. 2.5 Unequal Measurement Precision Applied to Regression Cook introduced this perturbation in his original (1986) paper on local influence, and the results in this section come from his work. An n x 1 vector of perturbations is introduced such that the perturbed likelihood is based upon y N(Xf3, U2D(1/w)), where D(1/w) denotes a diagonal matrix with ijh element -. The null perturbation is w0 = In. The n basic perturbations would be the local versions of the n Cook's distances (Cook 1977; Cook 1979). In addition, the n basic perturbations applied to LDIOA(w) are local influence analogs of the n DFBETAS for/3. A second interpretation of this perturbation scheme comes from the idea of case-weights. Case-weights are used in an analysis when each observation has been sampled with unequal probability. Under disproportionate sampling, cases are weighted by the inverse of their sampling probability in order to obtain unbiased regression parameter estimates. In the normal linear model, this weighting is numerically equivalent to perturbing the error variances. Hence, this perturbation scheme can be used to examine sensitivity to the assumption of random sampling. 2.5.1 Building Blocks To begin, we develop the perturbed likelihood, which is 2 1 L,,(8; y) oc -j log a2 2a--(y X/3'D~w)y X/3) nloga2 -_ 1 'D(w). 2 20r2 The next step is construction of A. Partial derivatives with respect to /3 are as follows: OL,(0; y) -1 (-2X'D(w)y + 2X'D(w)X)3) 913 2ar = X'D(w)p, and the partial derivative with respect to a2 is OL,,(O; y) n 1 02 2- + D(.rr ,9,T22 (40 v' ()y Second partial derivatives are then a2L(O; y) lX'D() 92L(O; y) y) 9a29I 24w and evaluating at the original model yields: 2L(O;y) = -X'D(r) 9039W' e=,=w &2 92L.(;y) 0; ,.yo 1 1 o2Ow' -,=0, 24 (r r)' = r'sq, where is the denotes the Hadamard (element-wise) product, implying that rsq is an n x 1 vector with elements r?. Combining these we can write A 2- D(r)X jrq and P=2&2 1D(r)X (XXy)-1 1X'D^) 454 [[( ) +4 rsqrf (2.15) _n [ 1 07 8 q 2 [D(r)HD(r) + 1 r..r P 2n2 r sqr . 2.5.2 Directed Influence on 62 The acceleration matrix for the profile likelihood displacement for &2 is simply the second term of (2.15): W 4 1rqrq (2.16) Solving the characteristic equations directly, the single eigensolution isn Solving the characteristic equations directly, the single eigensolution is Cmax 4 ma Sn&4 vmax oc rsq This suggests that &2 is most affected by variance perturbations when they are scaled according to the squared residuals. 2.5.3 Directed influence on ij Without loss of generality, consider directed influence on /31. Again we will proceed directly to determining the sole eigenvector proportional to A'T and the corresponding curvature 211A'T|12 as per (2.3) and (2.9): Vmiax (X D(r) X[ X (1) 17 S1D(r) [x1 X(1)Yi] (2.17) 1 = --D(r)rl &Jlr1JJ ox r rl and S=2n 211 r._ 12=I22. (2.18) cm 1J I I1 rJ11 r=1 This suggests that maximum local influence of the perturbation on /i is achieved by perturbing cases that have both large residuals and extreme values for the adjusted covariate. Cook (1986) provided additional details of how this result compares to the information in a detrended added-variable plot. Also, we may compare the information in Vma. with that provided by the DFBETAS diagnostic (Belsley et al. 1980). Recall that the value of DFBETASi,j is the standardized difference between /3j and j,(i), the estimate when deleting the ith observation. The formula is given by SDFBETAS 3j,(i) c- jri, DFBETALV J r Vr(j) scjl(1 hI ) where r, is the ith residual, cj is the jh column of X(X'X)-', cij is the ith element of cj, and s2 = 12 is the mean squared-error estimate of a2. The vector c, can be n written in terms of the jth adjusted regressor by partitioning (X'X)-1 as per (2.7). Forj = 1, [- 1 1 1 Ci X, X(1) ]f --I [P r]. Using this equality, the vector of n DFBETAS for /3j is proportional to 1 1-hil 1 1-h22 rr. (2.19) 1 l-hnn Since Vma x ocr rj, observations with large leverage stand out more in a plot of the n DFBETAS for fj than in a plot of Vmax,. This algebraic relationship between Vma and DFBETAS has not been noted by previous authors. 2.5.4 Directed influence on/ The acceleration matrix for directed influence on 3 is the first term of (2.15): = j [D(r)HD(r)]. The eigensystem of the matrix does not have a closed-form. However, some idea of the directed influence of this perturbation on 3 can be ascertained by utilizing basic perturbations. The ith basic perturbation corresponds to modifying the variance of only the ith case, and its curvature is the ith diagonal element of F'1 (Cook 1986): Cb = r2 hii. (2.20) (7U 1 1 0.75 0.75 0.5 0.5 0.25 0.25 6 S 4 0y $@il 0 a-- - 5. ."- '" 2 3-0 5" "'is 2 5 -0 -' 35 -0.25 -0.25 -0.5 -0.5 -0.75 -0,75 -1 -1 (a) V2 for (b) v3 for Figure 2.1: Hills data; variance perturbation This is similar to the ih Cook's Distance (Cook 1977): 1 2h,. k(1 hi)2s2 It, 2.5.5 Influence on 0 The eigensystem of F is not closed-form, but it can be found numerically. As an example, consider the perturbation of the error variances for the Scottish hills data. There are three non-zero eigenvalues for F: 14.82, 4.91, and 0.37. The eigenvector associated with the largest eigenvalue was plotted in Figure 1.10(a). It is similar to Vma.,, for directed influence on &2, as given in Figure 1.10(b). The other two eigenvectors for F are plotted in Figure 2.1. 2.6 Benchmarks for Perturbation Analyses An analyst using local influence diagnostics will want to know when a curvature is "too big" and when an element of Vmax is "too big." Calibration of local influence diagnostics was the topic of much debate in the discussion and rejoinder to Cook (1986), and there is still no consensus opinion on the subject. Much of the difficulty lies in the fact that the perturbation is introduced into the problem by the analyst. Because the technique does not involve hypothesis testing, critical values are not appropriate. However, there have been attempts to develop rule-of-thumb benchmarks, as exist for Cook's Distance and other common influence diagnostics. In this section, we discuss some general difficulties in developing benchmarks for perturbation analyses, as well as some issues specific to the local influence approach. Although it is argued that all sensitivity is "relative", some simple methods for assessing the size of curvatures and the size of the elements of directional vectors are presented. 2.6.1 The Analyst's Dilemma As mentioned before, the perturbation is introduced by the analyst, and this implies that he/she must decide whether the effects of perturbation are important, much the way a researcher decides what effect size is important. Consider a situation in which the researcher is actually able to give the size or stochastic behavior of the perturbation. Here, we may concretely measure the effects of perturbation by whether or not the conclusion of a hypothesis test changes. Alternatively, the effects can be measured by whether or not the perturbed estimates are outside an unperturbed confidence ellipsoid. However, despite having a concrete measure of the perturbations's effects, assessing the practical relevance of the results still falls to the analyst. 2.6.2 Calibrating Curvatures in Local Influence In addition to the general limitations of a perturbation analysis, an analyst using the local influence approach must also contend with the fact that the behavior and size of the perturbations are undetermined apart from the basic formulation. Loynes (1986) and Beckman et al. (1987) noted that curvatures do not have the invariance properties needed for a catch-all method of benchmarking. This necessitates a reliance upon relative influence. 2.6.2.1 1-to-1 coordinate changes in fl Following Loynes (1986) and Beckman et al. (1987), let us consider 1-to-1 coordinate changes in the perturbation space f. That is, a new perturbation scheme is constructed with elements w! = k(wa) for i = 1,... q, for a smooth function k. The new scheme's perturbation, w*, will not yield the same curvature as w. Specifically, C. = (q _2 C,, where w,, denotes the null perturbation value for wi. Ow=w0 Wo -2 G2 As an example, consider perturbing the variance of y1 to be 2 rather than 1-. Here, k(w1) = w?, resulting in new curvatures that are (_01? .i)2 (2w ) =1 = 4 times the original ones. Hence, despite the fact that the two versions of 0 can produce the same set of perturbed models, the curvature in identical directions will change. This implies that a universal benchmark for curvatures is not possible. Indeed, given the arbitrariness of scaling the perturbation, it should come as no surprise that curvatures cannot be compared across perturbation schemes. However, the direction of maximum curvature is unchanged under 1-to-1 coordinate changes in fl. 2.6.2.2 Interpreting curvatures In this section, a simple way to assess curvatures from the same perturbation 2 scheme is given. First, RC is a Taylor series approximation to LD(wo + av) 2 (Lawrance 1991; Escobar and Meeker 1992). Also, assuming that n is large, an asymptotic 100(1 a)% likelihood-based confidence region for 0 is given by values of 0 such that 2(L(; y)- L(O;y)) < Xp,-. Thus, if twice the curvature in direction v is larger than 2x,1-, then a perturbation at a distance of one from w0 in direction v moves the parameter estimates to the edge of the asymptotic confidence region. To further develop this idea, let us define the size of a perturbation to be its Euclidean distance from the null perturbation (i.e., 11w Woll). Suppose further that we have perturbations in two different directions: WA in direction VA and WB in direction VB. If ICA\ is k times larger than IC\I, then perturbation WB must be vk times larger than WA to have the same effect on the likelihood displacement. Equivalently, in order to perturb the estimates to the edge of the confidence ellipsoid, it would be necessary to travel Vk times farther in direction VB in 0 as in direction VA. This provides a simple interpretation for the relative size of two curvatures. 2.6.2.3 Internal benchmarks for curvatures Building on the interpretation given above, curvatures can be benchmarked based on their relative size to some sort of "average" curvature for a given scheme and fitted model. For instance, the average of the q basic curvatures for the fitted model can serve as a benchmark. Alternatively, the average curvature could be computed by (1) allowing the squared elements of v to have a Dirichlet(al ... aq) distribution with all ai equal to one, and then (2) finding the expected curvature. This would be equivalent to finding the average approximate displacement at a radius of V2i from Wo (Cook 1986). If such a benchmark is difficult to compute, a Monte Carlo estimate can be calculated. Because these benchmarks are computed using curvatures from the fitted model, they result in comparisons that are internal to the fitted model. The conformal normal curvature of Poon and Poon (1999) would also be an internal way to assess the sizes of curvatures. 2.6.2.4 External benchmarks for curvatures It is entirely possible that all of the curvatures of a fitted model are large due to the nature of the data set, and using internal benchmarks will not alert the researcher to such a phenomenon. To address this, a benchmark that is not dependent on the observed data is needed. To this end, we might utilize the expected value of the curvature in direction v, using 0 in place of the unknown 40 parameters 0. The expected curvatures can then be plotted along with the observed curvatures (Cadigan and Farrell 1999). If the expectation of C.ma is difficult to obtain, Monte Carlo simulations could be used to estimate it. As an alternative, Schall and Dunne (1992) developed a means of external benchmarking by developing a scaled curvature using results on parameter orthogonality (Cox and Reid 1987). 2.6.2.5 Internal versus external benchmarks Internal and external benchmarks are complimentary tools, since each one provides different diagnostic information. Comparing curvatures from the same fitted model highlights relative sensitivity given the data and the design. Comparing curvatures to their expectations highlights whether the sensitivities are expected given the experimental or sampling design. 2.6.3 Calibration of Directional Vectors For a directional vector from the eigensystem of F, the analyst must compare the individual elements in the vector to determine which aspects of the perturbation are most important. Typically, it is suggested that the analyst simply look for elements that stand out as being large relative to the rest. Alternately, a vector whose elements are all equal in size can serve as a means of comparison. That is to say, if each of the q elements of the perturbation are contributing equally, then each is equal to either or --. A particular element whose absolute value is larger than, say, 2 or 3 times that of equally contributing elements (e.c.e.) would then be of interest. Hence, the inclusion of horizontal lines at 2- and -- in the plot of directional vectors may be useful. 2.7 Local Assessment Under Reparameterizations Pan et al. (1997) show that Vmax is invariant to any 1-to-1 measureable transformation of the parameter space R. In fact, for 0 = h(k), the entire eigensystem of Fk, is the same as that of F0. However, directed influence on the estimates of the redefined parameters can then be examined. Using similar steps to Pan et al. (1997) and Escobar and Meeker (1992), we can write update formulas for Ao and Lo and then calculate directed influence on a subset of the redefined parameter estimates. Using partial differentiation, the building blocks of the acceleration matrix under the parameterization 0 can be written: 00' A o= TA (2.21) A = (-Aoi a I -o- -) a(2.22) (900 -' -1 1(00'1 'got ) LO (-50-, where all partial derivatives are evaluated at the fitted model. From this, the acceleration matrix for directed influence on 01 can be obtained from the partition 0 = (01, 02)': F1 -2 = -2A,(L-1 B02)A- = 00...i 00' 2A' (L B22)a0-AO (2.23) 00' -2A( (L- -B 2-)AO, where 0 0 B0^22- 0 L-1 B~22=[022 L22 comes from the partition Loll L 121 L021 L022 and all partial derivatives are again evaluated at the fitted model. Although Pan et al. (1997) suggested that (2.23) is invariant to the choice of 02, they did not provide a proof. To show this result, consider another reparameterization for which the first pi elements are the same as those of 4: 2 02 This implies that = I P 0 ((2.24) and this leads to the following expression for the lower right submatrix of L: o ao' .. ao a2 where all partial derivatives are evaluated at the original model. Meanwhile, the acceleration matrix for directed influence on i/1 is Loo a -2A O a1' l-1 9 *"[ 122 ,i- ~ ~ i o V 2 a0(2.26) t9_..-1 00 0 a0 L022 - aoq,) 1 90f2 f 2 where all partial derivatives are evaluated at the original model. Meanwhile, the The invariance of the acceleration matrix for directed influence on to the choices of 2 is proved by showing that (2.26) is equivalent to (2.23). Using the inverse of 0 L-1 102 (2.26) ao i 9 0 0 ao, =-2A(L -9 )A9. 0 "22 The invariance of the acceleration matrix for directed influence on 01> to the choice Of 0^2 is proved by showing that (2.26) is equivalent to (2.23). Using the inverse of (2.25), D0 al' 0 0 D0' 0 L-1 9D IP22 Substituting (2.27) into oo 4o 0 1 Dk'.9o' DO [9 o 0 0 90l 9O' 0 L-1022 L90 O4 0 0 0 02 002 a l DO0 0 0 O0' 9) lo 9 .2LT-1 L-1 Q0D L 0 -' 22 0 22 2 DO 0 0 1O' 90' o L-1 90 (2.26) completes the result. (2.26) completes the result. (2.27) 2.7.1 Directed Influence on a Linear Combination of Parameter Estimates As mentioned in Escobar and Meeker (1992), a specific application of these results is to consider directed influence on a linear combination of parameter estimates, x'O. There are many ways to reparameterize the model such that x'O is one of the new parameters. For example, assuming that the first element of x is non-zero, 40 can be defined as AO, where A zX(1) oP_1 IP_1 where x, and x(i) are the first and remaining elements of x, respectively (Escobar and Meeker 1992). Using equation (2.23), the acceleration matrix for directed influence on 01 = x'O can be computed, and the resulting diagnostics are invariant to the choice of 02, .., Op. By using a prudent choice of new parameters, we can obtain easy-to-compute formulas for the single eigensolution of F [], as per the following theorem. 44 Theorem 2.1 The maximum curvature and its direction for directed influence on z'O is given by Cmax = lk2 1IA0LOI2 'max "'-1 X "-1/2 where ki = L/2x. Proof of Theorem 2.1 To begin, consider a model with parameterization K' = K'l2 0, where K ki k2 ... kp] is a full rank p x p matrix with Ic -1/ ki = l/2x kIkj= 0 i/j k'iki kkl i= 2...p. By construction, 1 S= X'0 K-1 K' Ilk,112 000 0 1/2 Ko (90 1 LO1K. Using the update formulas in (2.21) and (2.22) it follows that A 1- 10 A4 = -9-- (5- 707) L L(1=O0 -ILlj, 1(0-- K,11/2ilil/2K = Ilk, 112j_ L 0 0 Next, we show that directed influence on ^ = x'O is not a function of k2,..., kp. From equation (2.23), we find x'"-]- 2( Ao i1/2K) (lkl121- lkl112 [ ] ( Klil2Ae) 2 A Loi/2A K 1 ] K'L91'2A kllk,2 0 00 2 A' iL-1/2 kkl I1/2 2 illill -lk,112 AoLe xz LO Ao. (2.28) The single eigensolution of this rank-one matrix yields the final solution for Cmax and Vmax. E The construction of 4 is worth some note. The premultiplication of 0 by K'Lf/2 orthogonalizes the parameters, in the sense that the off-diagonal elements of the observed information matrix for 0 are zero. (Note that the orthogonalization is with respect to the observed information matrix as in Tawn (1987), rather "1/20 than the expected information as in Cox and Reid (1987).) First, a = Le20 is a parameterization with an observed information matrix equal to the identity matrix. Then, K is constructed to make orthogonal contrasts of a, with the first column corresponding to the linear combination of interest. Finally, by scaling the columns of K to all have the same length as ki, K becomes a scalar multiple of an orthonormal matrix, and this leads to the algebraic simplifications. Also, if x'0 = 0j the local influence diagnostics provided by the theorem are the same as those provided by (2.3), as per Beckman et al. (1987). 2.7.2 Application to Regression As an example, consider perturbing the variances of a regression model and examining influence on a linear combination of the regression parameter estimates, A(x) = x'f3. From (2.28), we find F iL 1 I (2.29) = -- D(rX(X'X)-'xx'(X'X)-IX'D(r), W2h, where h. = x'(X'X)-lx Further, Vma c x'(X'X)-1X'D(r) oc Cov(X/3,x'3) -r (2.30) Cma = 2ix'(X'X)-X'D(r)2 (2.31) &2h. where Cov(Xo, x'/3) is the vector of empirical covariances between Xf3 and x'f3. 2.7.2.1 A DFFITS-type measure Using the variance perturbation and directed influence on z'/3, a local influence analog of DFFITS can be defined. Recall that DFFITS. is defined as the externally studentized change in the ith fitted value when deleting the ith observation (Belsley et al. 1980): DFFITS. = - r i]hisi where p(xi)[ij is the estimate of p(xi) with the ith observation deleted and s22 is the [i] mean squared-error estimate of a2 with the ih observation deleted. A large value for DFFITS indicates that the observation is somewhat "self-predicting". It can be shown that DFFITS, simplifies to (1 /h i,) [i] A similar measure can be obtained by examining local influence on p2(xi) = xif/ when using the ith basic variance perturbation. That is, only the ih observation is reweighted, and the curvature for directed influence on f(xi) is examined. The curvature for the ith basic perturbation is the ith diagonal element of 2,r~hii (.2 Cb, = BasicFITSi = 2r (2.32) A large value of BasicFITS indicates that the perturbation of the observation's precision/weighting greatly influences its fitted value. Thus it is a local influence analog of DFFITS. Coincidently, BasicFITS5 is equivalent to the ith basic curvature for the acceleration matrix, as given in (2.20). That implies that for the variance perturbation, the n basic curvatures of LD(w) are measuring influence on the n fitted values. While BasicFITS, is the ith basic curvature when directing influence on /(xi), we can also define MaxFITS, to be Cma, for directed influence on fA(xi). Using (2.31), Cmaxi = MaxFITS. 2 r h "(2.33) 2h Ziihj= By definition the, ith MaxFITS can be no smaller than the ith BasicFITS. While BasicFITS provides information about which observation's perturbation influences its fitted value, MaxFITS provides information about how sensitive each fitted value is relative to the others. Example Consider examining influence on the fitted values for each of the 35 covariate profiles in the Scottish hills data set. Figure 2.2(a) displays the 35 BasicFITS (i.e., each of the basic curvatures for directed influence on the corresponding ft(xi)) . Race 7 is indicated as being influential on its fitted value, while race 18 is also shown * 5 10 15 20 25 30 35 (a) BasicFITS 0 S *0 I 0 0 6 *0 * 00 0 * 00 40 5 10 15 20 25 30 35 (b) MaxFITS (c) BasicFITS (stars connected by solid line) and MaxFITS (triangles connected by dashed line) Figure 2.2: Hills data; a2D(1/w) perturbation; influence on fitted values to be quite influential on its fitted value. Plot (b) shows the 35 MaxFITS (i.e. the Cmax'S for directed influence on each of the 35 /(xi)). Since no point stands out as being large, the plot indicates that no fitted value is particularly sensitivity to variance perturbations. Plot (c) shows both sets of diagnostics. The BasicFITS values are stars joined by a solid line, while the MaxFITS values are triangles joined by a dashed line. The closeness of the MaxFITS and BasicFITS values for race 7 indicates that this observation is somewhat self-predicting. CHAPTER 3 PERTURBATIONS TO THE X MATRIX IN REGRESSION In this chapter, the influence that infinitesimal errors in the X matrix can have on parameter estimates in a linear regression is addressed. First, previous results for single column and single row perturbations are reviewed. This is followed by a generalization of these results to perturbing all elements in the X matrix. It is shown that the eigensolution of P[ is related to principal components regression, collinearity, and outliers. Finally, new graphical techniques are presented with the results applied to the Scottish hills data set and an educational outcomes data set. 3.1 Perturbations to Explanatory Variables Letting k denote the number of predictors, Cook (1986) proposed that an nk-sized perturbation can be incorporated into the design matrix in an additive fashion. These perturbations represent fixed errors in the regressors and should not be confused with random errors-in-variables techniques (Fuller 1987). The perturbation scheme can be used to assess the effects of small rounding or truncation errors in the explanatory variables. Upon perturbation, the X matrix becomes W1 Wn+l W2n+l ... Wkn-n+1 W2 Wn+2 ... ... Wkn-n+2 X + W3 ... ... ... Wkn-n+3 Wn W2n W3n ... WCkn The use of a single subscript implies w = vec(W), where the vec operator stacks the columns of the matrix into a single column. When particular elements of W greatly affect parameter estimation, it indicates that the corresponding element of X is influential. This allows the analyst to identify specific combinations of explanatory variable values that influence estimation. As will be shown in Section 3.4, sensitivity of the regression parameter estimates to these kinds of perturbations is greater when there is collinearity amongst the regressors and when the model is calibrated to fit the observed data too closely. 3.2 Perturbations to a Single Column The special case of only perturbing a single column of X was discussed in Cook (1986), Thomas and Cook (1989), and Cook and Weisberg (1991). This section recreates their results in detail. 3.2.1 Building Blocks Here and in the following sections, di represents a vector with a 1 in the ith position and zeros elsewhere. The length of the vector depends on the context of its use. Note that for di of size k, 1 d(X'X)- d i -rI2 (3.1) and 1 X(X'X)-ld, = 1, ,r, (3.2) where r, contains the residuals from regressing X, on the other columns of X. Without loss of generality, suppose perturbations are added to the n values of the first predictor (i.e., X1 becomes X1 + w). The perturbed likelihood is L(; y) oc -n log2 21 (- X'3 A)'y- XP3- '1w) 2 2 (3.3) =- _logu2 1 2W + 'w). O2 -o + 0 51 The next step is the construction of A. Partial derivatives with respect to w are as follows: 9L,(O; y) (w =- (y- X 13W). 0- (3.4) Second partial derivatives are then 02L,(0O; y) 9w9/31 02L (0; y) 9w 013(l 02L.(O; y) OwaO2 1 =^(y-x(1)&l)- 2/3i(x,+u,)) - -(y-Xf 1W). O04 Evaluating at the original model yields i2Lw(O; y) 0w0fa 0=0,W=Wo 02L,(0; y) 09u90l() O=,w=wo 02L,(0; y) 0wQ^a2 e=eW 1 1 (r 4lX,) = -x(1) /41 a4, and from these quantities we construct the n x p matrix A' i[r-/l- =- rd' lX -]X(1) -Jr] rI ] where d, is a k x 1 vector. It follows that the acceleration matrix is given by (3.5) dlr' -- 1x r/ P 21 rd -1 'xx')-i o ] 2 rd _-lx -I] ^01 2&--4 n =2 + 2- ) rrI + 121H -l(rrl + r)r') 3.2.2 Directed Influence on &2 The acceleration matrix for the profile likelihood displacement for &2 is given by F[] = A^,2A (3.6) n ,rr. (3.7) Solving the characteristic equations (F AI)v = 0 directly, the single eigensolution is Cmax a v max cX r. Therefore the direction of maximum curvature for directed influence on &2 is always proportional to the residuals regardless of which column of X is perturbed, but the corresponding curvature is a dependent upon the effect size, |/3i|. 3.2.3 Directed Influence on/ i Using the results of Beckman et al. (1987), the sole eigenvector is proportional to A'T and the corresponding curvature is 2|IA'TI12 as per (2.3): 1 [ ] 5 a1 Vmax 0C r 7 /41X1 -1-X(i) &1[ ] 1 [r 431Xl + 1 1 X] (3.8) &I1ri|l Ic r 4r r and Cmax 2 r- il' = 211= 2 1 2'- /1, 112 & 2^jp [rr 2 _ri'+ +4172 (3.9) =2 1( + Ir2 The direction of maximum curvature is a weighted average of the residuals and the adjusted regressor. As noted in Cook and Weisberg (1991), 41r1 = y y(), where Y() is the vector of fitted values when X1 is excluded from the model. Using this result, we find Vma, oc 2r rl), where r*) is the vector of residuals when modeling y using all regressors except X1. This indicates that the direction is dominated by the residuals, a fact which is demonstrated by examples later in this chapter. Cook and Weisberg (1991) discussed how Vmax relates to de-trended added variable plots. 3.2.4 Directed Influence on Other Regression Parameter Estimates Thomas and Cook (1989) examined the influence that perturbations in one column of X can have on the regression parameter estimate corresponding to a different column. Without loss of generality, suppose we perturb the first column of X and look at the effects on the last regression parameter estimate /k. Again using the results of Beckman et al. (1987) we can obtain the maximum curvature and its direction. Letting di be of size k 1, vm, oc AT [ (rd, ,X(,)) ] X -f -i rkI [-r k + ,-H(k)Xk IlXkL (3.10) ocC ckir Ilrk, where ikl is the regression parameter estimate corresponding to X1 when predicting Xk with the other columns of the X matrix. The maximum curvature is given by Cmax = [i ,2kir'+ ir'k] [kir +OArJ S icTrk-- k [ 1 + / (3.11) 2 + II- H2 . 1+ Y1 Irk112! When the partial correlation between X1 and Xk is zero, Cmax achieves its minimum value of -, yielding Vmax cx rk- In the event that Xk is a linear combination of the other regressors with non-zero ykl, Cmax = oo and Vmax c r. Between these two extemes, the direction of maximum curvature is a weighted average of the residuals and the adjusted kth regressor. These quantities are used to detect outliers and measure leverage, respectively. If the partial correlation between X1 and Xk is as large as that between X1 and y, then the two vectors are equally weighted. Note that this weighting is invariant to scale changes in the columns of the design matrix as well as the response. 3.2.4.1 Directed influence on / Without loss of generality, we again consider perturbing the first column. The rank-k acceleration matrix is =-- 2 ( 1 rr' + H- (-rr'i + rr') (3.12) The first eigenvalue of this matrix is unique, and yields a maximum curvature and corresponding direction of Cmax = (A2 + Jr ) and vi oc r- lrl. Hence, the maximum curvature and its direction are identical to that of directed influence on i. Confirmation of this first eigensolution follows: -A11) vi (" 4' (i2 + HPE) I) (r _ oc (rr' + _riII2IH I,(rr[ + ,r')) (r ihi) -(/32r 112 + 11712j1( ~~ -( ,ll+IIl (r ri) = Itr112r &I 112lri IIl'lII1Hr', + K1r, 112 r rllr llr 111rr + Illri ll2r, + llrll2, = -kl,.,lt112Hr, + 'Jll,.11l ri =0. The second eigenvalue is /321 and has multiplicity k 1. As mentioned earlier, this implies that there is no unique set of orthogonal eigenvectors to complete the solution. 3.2.5 Influence on The acceleration matrix for examining both f3 and &2 simulatenously has k + 1 = p non-zero eigenvalues. The form of the eigensolution is unknown. My numerical studies indicate that there are three unique non-zero eigenvalues, one of which has multiplicity k 1. The one with higher multiplicity appears to be the second largest of the three, with a value of'7. ^2 Example Huynh (1982) and Fung and Kwan (1997) examined data from 20 randomly sampled schools from the Mid-Atlantic and New England States. The mean verbal test score (VSCORE) of each school's 6th graders is regressed on staff salaries per pupil (SALARY), % of white-collar fathers (WCOLR), a socio-economic composite score (SES), mean teacher's verbal score (TSCORE), and mean mother's educational Table 3.1: School data; Pearson correlation coefficients SALARY WCOLR SES TSCORE MOMED VSCORE SALARY 1.00 WCOLR 0.18 1.00 SES 0.22 0.82 1.00 TSCORE 0.50 0.05 0.18 1.00 MOMED 0.19 0.92 0.81 0.12 1.00 VSCORE 0.19 0.75 0.92 0.33 0.73 1.00 level (MOMED). Appendix A contains model fitting information and diagnostics for the recession analysis, with all variables centered and scaled. Diagnostic plots of the residuals, externally studentized residuals, Cook's Distances, diagonal elements of the hat matrix, and DFBETAS appear in Figure 3.1. Ordinary least squares regression indicates cases 3 and 18 are outliers, although Huynh (1982) also identified observation 11 as a mild outlier. This data set also has some redundancy among the predictors SES, WCOLR and MOMED (Table 3.1). Although the X matrix is not ill-conditioned, the association between SES and MOMED accounts for the unexpected negative slope estimate for MOMED. Table 3.2 shows the non-zero curvatures (eigenvalues) for perturbing single columns of the X matrix. Comparing the columns of the table provides information on which regressor's measurement is most influential. Here we see that the measurement of SES is typically the most influential, both for regression parameter estimates and for &2. Comparing the rows provides information on which parameter estimates are most influenced by these kinds of perturbations. Here we see that WCOLR and MOMED estimates are the most sensitive. These are the two regressors that are highly correlated with each other and with SES. It comes as no surprise that the results for collinear explanatory variables are more susceptable to measurement errors of this nature. 9 9 I 9 ' * *S D *1 (a) Residuals S 10 1iS 20 (d) Hat diagonals I 0.5 9 *D.* -0. II 11 20 .1 d.! (g) DFBETAS for SES * t 9s 1 o 10 20 iz 5 "'* U "1'S ' e . (b) Externally stu- dentized residuals 9 9 *~ -- *.*. * ~ t~ .91b (e) DFBETAS for SALARY 999 9 10 ~ 11 l.1~ (h) DFBETAS for TSCORE 0.3 0.9 ..... .. .. 'D .' o 5 It is 20 (c) Cook's Dis- tances Mi 1.5 # 1. 0.51 .9 *, 9 "9 9 0.5 0.1 (f) DFBETAS for WCOLR (i) DFBETAS for MOMED Figure 3.1: School data; diagnostic plots * 0 , . '. ,.'IV 'fO Table 3.2: School data; curvatures using X, + w perturbation Estimates Column being perturbed in LD(w) or LD[el](w) SALARY WCOLR SES TSCORE MOMED 0 A1=4.144 A1=20.156 A1=50.008 A1=6.658 A1=19.065 A2=0.440 A2=0.847 A2=19.002 A2=1.413 A2=00.932 A3=0.094 A3=0.071 A3=14.441 A3=0.599 A3=00.091 3 A1=3.357 A1=18.534 A1=26.445 A1=4.432 A1=17.292 A2=0.440 A2=0.847 A2=19.002 A2=1.413 A2=00.932 i 3.357 0.956 19.008 2.126 0.951 /2 0.458 18.534 19.890 1.585 10.869 3 0.443 2.957 26.445 1.501 1.640 /4 1.129 1.859 19.220 4.432 1.295 /5 0.444 11.590 19.324 1.480 17.292 &2 0.881 1.693 38.004 2.825 1.865 Since perturbing the SES variable is most influential, examining directional vectors for this perturbation is of interest. The element corresponding to case 18 is the most noticeable in the plot of vmax for the likelihood displacement (Figure 3.2). This observation has a large residual and is indicated as being influential in the various diagnostic plots in Figure 3.1. Also note the similarity of Figure 3.2 to the residuals (Figure 3.1(a)). Plots (a)-(e) of Figure 3.3 show the directions of maximum curvature when directing influence on each of the individual regression parameter estimates. In general the plots indicate different elements of X3 to be influential on different /3. Finally, Figure 3.3(f) provides the direction of maximum curvature when directing influence on &2. Recall that this vector is proportional to the residuals. 1 0.75 0.5 0.25 0 0 .5o ft 15 20 o o -0.25 -0.5 * -0.75 -1 Figure 3.2: School data; X3 + w perturbation; influence on/3 3.3 Perturbations to a Single Row The special case of only perturbing a single row of X was discussed in Thomas and Cook (1989) in generalized linear models. Here, detailed derivations for regression are presented. The results indicate that single row perturbations have limited diagnostic value. 3.3.1 Building Blocks Without loss of generality, suppose perturbations are added to the k regressor values of the first observation (i.e., x' becomes x' + w'). For d, being of order n, the perturbed likelihood is L (O;y) oc log ,2 -1 (y X3 d,'f3'(y X dlw,') 22a 2 1 k k (3.13) = -log (' 2 y- 2( Owj) + ( jWj)2). j=1 j=l Partial derivatives with respect to w are as follows: '2 (3.(-214+2()j) 9w 212 (3.14) =1-CE m 0, S 0 5 0 0 1- - -1 .* .41 I 1 '10 n 20 o -0.25 S-0.5 *0.75 *I (a) vma. for di 1 0.75 S 0.5 o 0.25 15 20 .5 7 2 -0.25 .- (c) vma for /3 * O000. 15 w 1.g I a O **0 5 10 15 020 # I (b) Vmx for f2 O* O I O S * 5 o 10 O N .,ioA (d) Vmax for 04 0 O 5 0 10 O * o15 -o -- o o (e) Vmax for /35 (f) Vmax for &2 Figure 3.3: School data; X3 + w perturbation; directed influence plots o Oo B 5 * I Since the null perturbation is 0, evaluating at w0 simplifies the expression. We then proceed to take second partial derivatives: 92L,(0; y) 1 4W49,8i L O a2 0 0 Yi x'f1 0 0 &2L.(0; y) (Y_ -x[03. aJ9(:r2 W=Oo = ( O4 Evaluating at 0 = 0 and combining the portions for the regression parameter estimates we obtain where r, is the first matrix a2L .( 0; y) 1 l 1 0wao' 10=,w o Oxi 02 L,,(0; y) 8=,=o r, L(r2;y) T- 4 residual. From these quantities we can construct the k x p A'=I (3.15) The acceleration matrix is then given by S. 2 r2 I- 1 [ (OP1 2 I 4 r- ' n hr - = 2 [(hil + 2r2) 8'r 2 (X'X)-' r1Mx (X'X)-1 ri(X'X)-YXi'] T2 ~~~11rJ21 - Xl0 3.3.2 Directed Influence on 62 The acceleration matrix for the profile likelihood displacement for &2 is given by F[J-o ^Q2 [2&14] [2 r n &4J (3.16) n4r Solving the characteristic equations (F2] AI)v directly, the single eigensolution is Ca= ^IIn VmaxOC. Notice how the curvature depends upon the residual for the case that is being perturbed, but the directional vector is always proportional to 3 regardless of which row of X is being perturbed. This suggests that Vmax is not very useful for single row perturbations. The values of Cmax for each row can be compared to assess which row is most influential when perturbed. However, the relative effects only depend upon the residuals, indicating there is little value added by plotting the curvatures. 3.3.3 Directed Influence on f3. Without loss of generality, we can direct influence on 01: Vmax oC AT C [rlIk[ 1 --1 riIk-1 -- I0 -X rl /-'- 1ll + (ill Xll)3 hl] 21=|( + -x-2, Cm.= p-- ,lrl II Y- *[)' + (:ll- X1:0011"2 where il is the predicted value of xll from modeling X1 as a linear combination of the other regressors. More generally, the maximum local influence on ,Bj by perturbing row i is Vmax = ri + (iij xij)O ( -14Y ) Cmx= 2-11-rj II 11(1 ') + (i xij)l2. If the columns of X have all been standardized, then each of the elements of 7 and of /3 is less than 1. This suggests that element i of Vmx is likely to be larger than the other elements. However, the signs of ri and (iij xjj,)/ can work to cancel each other out. Further interpretation is suppressed in lieu of the more general Ok S( rl Ok-i rl( 1 64 Cmax 40 35 30 25 20 15 10 oo o*0 S O 0 5 O i Row 5 10 15 20 Figure 3.4: School data; maximum curvatures for single row perturbations perturbation presented in Section 3.4. Note that the curvature and directional vector change if one or more of the columns of X are rescaled. 3.3.4 Influence on 3 and b The acceleration matrix for directed influence on /3 is p = -2 hj + rh(X'X)- rlfx(X'X)-1 r(X'X)-1X }. (3.17) The eigensystem of this matrix is unknown, but my numerical examples indicate that it has k distinct non-zero eigenvalues. The eigensystem of F for the full set of parameter estimates is also unknown. 3.3.5 Example The values of school 18's regressors are most influential when perturbing rows of X one at a time, as evidenced by a large maximum curvature (Figure 3.4). For perturbing row 18, vmax is nearly proportional to /3 (Figure 3.5). Note that although element 3 of Vmax appears quite large, it is actually less than the benchmark of 2 = 894. 65 i 0.75 0.5 0.25 2 3 4 5 -0.25 * -0.5 -0.75 -1 Figure 3.5: School data; x18 + W', vm for 0 3.4 Perturbations to the Entire Design Matrix At this point, the general case of perturbing all nk elements of the design matrix is considered. We use W to denote the perturbations in matrix form: W1 Wn+l W2n+1 .. W.kn-n+1 W2 Wn+2 ... ... Wk n-n+2 W W w3 ... ... ... Wkn-n+3 Wn W2n W3n ... Wkn and wj to denote the jth column of W. 3.4.1 Building Blocks The derivations in this section only require the presence of an intercept in the model (or centering of the response y). However, for ease of interpretation, we assume the columns of X have been centered and scaled. The perturbed likelihood is L,(O;y) oc log o2 (y- XO W )'(y X3 W3). (3.18) 2 o 20r2 66 Proceeding in a similar fashion to the derivations for perturbing a single column, partial derivatives with respect to w are as follows: aL,,(e;y) 1 wJ a2 01(y - 02(Y - X3) /3WI3 XO) 0wf2 Since the null perturbation is 0, evaluating at wo simplifies the expression: 9L,,(O;y) 1_ 9Ow W=o a2 1(y-xf3) 01(y- XJ8) 0k(y XO) We then proceed to take second partial derivatives and evaluate at F r 0 ... 0\ t 1 ~a2 92 Lw 0; y)0=0 9WO9,3, e0e0,uW=W 0 ... 0: /31 I2x &kX L \ ={(Ikr-O3X) or 92L,(0;y) 1 9w9Oa2 O=Oo,W=Wo - (3.19) (3.20) i3ir ^2r &3kr A k(y XJ3) AWO 14(.39r). From these quantities we can construct the np x p matrix A', as given by Cook (1986): A'= [Ikr- X (r)] (3.21) where denotes the Kronecker product (Searle 1982). The next step is to construct the acceleration matrix. Here we use the fact that (A ( B)(C D) = AC 0 BD, assuming comformability of the matrices. Noting that A = A 1 = 1 A, the acceleration matrix can be written as 2=((I r- 9X (X'X) (Ikr- X')+4 (I'rr' 4+ \n / \ (3.22) 3.4.2 Directed Influence on 62 The acceleration matrix for the profile likelihood displacement for a2 is the second term of (3.22): 4= Of rr'. (3.23) n&4 This matrix has a single non-zero eigenvalue, |2, with corresponding eigenvector oc /3 r. This solution is confirmed below: [4 4 -I)v oc 4n rr' T l21 ] (03 9 r) = 40'&rr_11 11].(0r) 4- 0 [ 1 2 1 I M I2 r 11 1 12. 1 r ) =0. This result implies maximum local influence on 62 is achieved by perturbations in each column that are proportional to the residuals scaled by the corresponding j3. 3.4.3 Directed Influence on (3 In this section, we derive the full eigensystem of ], where pp] = 2 (1, 0 r X) ((X'X)-1 0 r' -/3 (X'X)-X'). The following theorem provides the full eigensystem of F for the X + W perturbation in a regression analysis. Note that eigenvalues in the theorem appeared in Cook (1986), but the eigenvectors appear here for the first time. Theorem 3.1 Recall that W. is the jth eigenvector of X'X and that Xoj is the j1th principal component. Letting 6j denote the jth eigenvalue of X'X, the k non-zero eigenvalues and eigenvectors of F[ are S= 2(6n + 2) V a W_-j+ 0 r 3 9 Xpkij+_, (3.25) ) 2L_j+1 -&2 jO kj1 7 ~ kjl forj = 1,...,k. Proof of Theorem 3.1 Because the algebraic expressions are so long, the eigensolutions are confirmed in two steps. Noting that 61 and Okj+1 are the jth eigensolution of (X'X)-1, we S-j+1 have ] n 11013"12 (9 r) ( -2( 3j- z)^^ 2 [(1, (9 r 0 0X) ((X'X)-l (9 r' 2' (X'X)-1X')] (W, (9) r) T22 2([ + Iim3I2)(W r) 2 Ik (ik r X) (It ^ -j )] 2 J(le + II^H12)(Wj 0 r) P~ [(Ir-) (IrI2 P) jjl12lI2)~0) (.6 2 [117112 r - X X^)1 -pI 2 Il2+ i2)( 9 r) &2 ij T 6j 3326 =2 liiir ^ ^ li ( x )]. 2 l02 W 9 r 1 f ( 0 ( 9 X j The following matrix multiplication yields the same expression as (3.26): -= [(k 9 r X) ((X'X) -' (X'X) -') (3 X Xv?) &2 (llrli 2- 2(1 + 1012)(0 ) Xpj) 2 [(Ik9 r 0 X) (0- _|13||2Wj)] 2(1!E + 10112)0 Xyj) = 2 [m11{-i_ r + ( Xa)] 2 ( !l + IIii2)( Xp) &2 2 3 = 2-1,1 r) 1112O (9 X\)] (3.27) Using (3.26) and (3.27), confirmation of the jth eigensolution is immediate: AI 2= 2n-- + 111)1 (Wk-j+l 9 r) -26-j+l &2 -_[, 2( n +x 11112) 0( X k- (3.28) =0. Finally, for j : j*, the orthogonality of the solutions follows: (W, (9 r i3 (D X^p)'(j., (9 r (g Xvi.) = j, 9 r'r ', ( 9 ,jX'r w ,r'X j, + '3 x'XX j, = 0- 0- 0+f0 + =0. 0 3.4.3.1 Interpreting Theorem 3.1 The closed-form eigesolution in Theorem 3.1 gives insight into how slight changes to the observed explanatory variables affect inference. To interpret the theorem, consider maximum curvature, as per (3.25): Cmax = 2( + ). (3.29) The eigenvalues of X'X are traditionally used to assess the amount of collinearity in the regressors. If 8k is small, there is redundancy in the explanatory variables, and this causes the inversion of X'X to be unstable. If this inversion is unstable, so too are the regression parameter estimates, 3 = (X'X)- X'y. The first term in Cma,,x is large when 5k is small, giving further evidence that the estimates are more sensitive when collinearity is present. The second term of Cma, is also interesting. It can be considered a naive measure of goodness of fit; when the relationships between the explanatory variables and y are strong, |11&32 is large and &2 is small. Hence, "better fitting" models are more sensitive to perturbation. However, 11'3112/&2 typically grows whenever additional variables are added to the model, indicating more complicated models are also more sensitive to pertubation. In fact, HII13I2/&2 bears some similarity to R2: R SSeg IIX0ii2 IIX 112 SSy y n(&2+IIX + 12) The quantity R2 is the proportion of variance explained by the model, and serves as a measure of goodness-of-fit. However, R2 can be made artificially large by adding additional explanatory variables, even when they contribute very little to the explanatory value of the model. The quantity 11 1H2/&2 behaves similarly: additional variables decrease the denominator and usually increase the numerator. Of course, the scale of the regressors is a major factor in the size of |I1|12, indicating that scaling the regressors would be prudent for interpretation. So, we may interpret Cmax in the following way. The vector of parameter estimates is more sensitive when (a) there are collinear variables, (b) there are a large number of explanatory variables, and (c) the model closely fits the data. We are not surprised to see that the first two phenomenon are sources of sensitivity. However, the suggestion that better-fitting models are more sensitive to measurement error may at first be disconcerting. When viewed in light of Type I and Type II errors, this is not the case. Recall that a Type I error occurs when a test is wrongly declared significant, while a Type II error occurs when a test is wrongly declared insignificant. Generally, a Type I error is considered the worse of the two. So, if there are strong relationships between X and y, the formula for Cma,, implies this might be missed because of measurement error in the regressors. However, if there is a weak relationship, perturbations to X would not make the relationship appear strong. In this sense, the regression model is better protected against Type I errors (concluding there are strong relationships when there are none) than Type II errors (concluding there are weak relationships when in fact there are strong ones) when there is measurement error in the regressors. As a final comment on Cma, the term 111112/&2 may be better described as an indicator of a closely fitting model rather than a well fitting model. This suggests that if we construct complicated models to fit the nuances of a data set, the resulting parameter estimates are more sensitive to rounding and measurement errors in the explanatory variables. Now let us consider Vma as per Theorem 3.1: Vmax OC Wk 9 r 3 X( k. (3.30) This vector and the other eigenvectors of Fl provide detailed information about which elements of X most affect the stability of the MLEs. The expression for Vma, shows how outliers and leverage in the final principal component regressor determine precisely what kinds of mismeasurements have large local influence on f3. The directional vector can be used to scale W to achieve maximum local influence: Wmax 0C [9klr lXfk Vk2r 3XXok ... Vkkr &kXp ]. (3.31) The elements in the first column are a weighted average of the residuals (r) and the regressor values of the last principal component regressor (Xpk). The respective weights are the first original regressor's contribution to the final principal component (kl) and the negative of the first regression parameter estimate (-i). Alternatively, we can think of the elements in the first row as being a weighted average of the explanatory variables' contributions to the final principal component (W) and the negative of the parameter estimates (0'). The respective weights are the first residual (ri) and the negative of the first observation's value for the final principal component (-xlk). 3.4.3.2 A special case In order to examine in more detail how the perturbations given in (3.31) affect /, we consider a simple situation for which there are closed-forms for the perturbed parameter estimates. Let us assume that the response and a pair of regressors have been centered and scaled to have length one, and that the regressors are orthogonal. The implications of these simplifications are as follows: |iy||= I|X1 = |Ix211 1 r'r= 1-/1-A2 1 = xy 2 =x'2Y 61= 6 =1 = 12. The first two eigenvalues of F are the same: Cma.A, --l- A2 = 2(n- + 2n 1 )1 &2i Pi2 22 and we may choose either of the following vectors for Vma: r- iXl -lX2 -42-X1 r' 42-X2 Without loss of generality we choose the first one. Letting X" denote the perturbed design matrix, we have Xo = X + aW X=X+aWr-X -X] X+aIr 4X, (3.32) = ar + (1-aIai)Xi X2-a 2X1 Xl,.w X2,w ] for some real number a. Now, consider the effects of perturbation on 41. Assuming a no-intercept model is still fit after perturbation, the perturbed estimate is given by r-,i Y (3.33) ll 1 1 11 where ri, is the residuals from fitting X,,, as a function of X2,". The building blocks needed are as follows: 1 X H2, = X2,(X, X2,)X -1 a/2X)(X'2 a2X'1) 1 a22(X r,,, = (I H )X = (1 a/ acf2)Xi + cX2 + ar ri,,y = (1 a~l ac42)41 + 42 + a(l 412 /22) I|ri,,|2 = (1 a ac42)2 + c2 a2(1 2 2) where c = a.2(1-a$. It follows that 2+a2p 01, a ( I ac2)41 + c42 + a(l -1 2 -_ 2) (1 a: ac/32)2 + c2 + a2(1 -12 2) For the sake of comparison, we can also consider perturbing only the first column. Perturbation in the direction of Vmax when directing effects on /31 results in the following perturbed X matrix: X, = X + a*W* = X+a* Ir- IX 0 (3.34) = [ a*r + (1 a*I )Xi X2] as per (3.8). Here, the perturbed estimate is given by &W a ) + a* (1 -1- 2) (1a )2 + a'2(1 _/ 2_ 22)" In order for the two perturbations to be comparable, they should be equal in size. This is accomplished by stacking the columns of the perturbation matrices and making the resulting vectors equal length. In other words, the Froebenius norms of aW and a*W* should be equal. Letting a* = a provides the necessary adjustment. Now we can plot &,, for some selected values of /p and I2, considering values of a ranging from, say, -V2 to vf2-. This eliminates perturbation matrices with Froebenius norms that are larger than that of the unperturbed X matrix. Figure 3.6 shows plots of &, for four sets of parameter estimates. The solid lines represents the parameter estimate in the direction Vmax for the two-column perturbation, while the dashed line represents parameter estimates in the direction Vsmax for the one-column perturbation. All of the influence curves in plots (a) and (b) have a value of .2 under the null perturbation, i.e. when a = 0. Likewise, the curves in plots (c) and (d) are all equal to .5 at the null perturbation. In all plots we see that the single column perturbation does not have the same effect as the dual column perturbation. The discrepancy between their effects is larger when /32 is larger (plots (b) and (d)). 3.4.4 Influence in Principal Components Regression Consider the canonical form of the model, as given in Chapter 1: Y = Za + 6, (3.35) where Z = XV and a = p'/3. The following theorem proves that the direction of maximum curvature is the direction of maximumum directed influence on &k. Theorem 3.2 Let & denote the MLEs of a for model (3.35). The pairs Aj, vj given in Theorem 3.1 are the maximum curvature and direction of maximum curvature for directed influence on &k-j+1, j = 1,..., k. -1 0,5-' 0.5 1 (a) Influence curves for /3 = .2, /2 =.4 Bi -1 .-e:~ 0.5 1 (b) Influence curves for /l = .2, /32 = .7 B1 0.5 1 (c) Influence curves for/3i = .5, (d) Influence curves for /3 = .5, /2 = .4 2 = .7 Figure 3.6: Effects of single and double double column; dashed line- column 1 onl column perturbations on Q\. Solid line- .............. Proof of Theorem 3.2 We begin by obtaining update formulas for A and L1 as per Section 2.7: A" = A0/ A/ A^ =App (3.36) = T,(Ifcr-f3X)p = ( k 0 r - ,x )3 1 1= (W L r X) &2 (3.37) = D(6 = &2D(1/6). Next, we find maximum directed influence on dk-j+l using the results of Beckman et al. (1987): Vmax 0C AT _,Wk+l 0 0 1 r&i - oc &2 W^ r XV] N jk-j+1 0 0 oc Wk-j+l r (- 9 X k-j+l Cmax 2| 2 (IAaT _|j+l 2 )(-1 9 r' 0 k-j+1X')(Wk-j+1 0 /3 0 X kj+l) 2 (n&2 1111 blI2k 3+l) &26k_j+l 2( 6k +1 _k-j+1 ar2 Since these expressions are the jth eigensolution of FP the proof is complete. 0 This theorem gives added insight into the direction of maximum curvature. In short, Vmax is the direction that changes &k the most. Similarly, the jth largest eigenvalue and associated eigenvector specifically correspond to effects on &k-j+l. Also, the sizes of the curvatures show that the estimated coefficients for the most important principal component regressors are also the least sensitive. Example 1 Consider the Scottish hills data set described in Chapter 1. The purpose of this example is twofold- to demonstrate how the X-matrix results can be displayed graphically, and to demonstrate how collinearity affects the curvature values when examining influence on f3. Figure 3.7 gives the direction of maximum curvature for directed influence on a2. The first 35 elements correspond to perturbations of the DISTANCE regressor values, while elements 36-70 correspond to perturbations of the CLIMB regressor values. The value for race 18's DISTANCE is the largest contributor to this influential perturbation. Elements 7 (DISTANCE for race 7) and 53 (CLIMB for race 18) are also large The information in Figure 3.7 can be difficult to discern because the perturbation vector enters into the likelihood in the form of a matrix, W. Another way to display Vma is to first take its absolute value and then to consider the elements corresponding to each column of W. Each portion can then be plotted against indices 1,...,n and shown together (Figure 3.8(a)). This graphical display provides a "profile plot" of each column of W's contribution to the direction of maximum curvature. Here we see again that elements corresponding to observations 18 and 7 as being the most influential. There is also some evidence that measurement error in the DISTANCE values (solid line) are more influential than error in the CLIMB values (dashed line). Figures 3.8(b) and 3.8(b) provide profile plots of the absolute value of the two directional vectors corresponding to v, and v2 of F Il. The first plot draws attention to observations 7, 11, and 18, while no observations appear noteworthy in the second plot. At this point we turn our attention to the curvature values that accompany these directional vectors (Table 3.3). The first column shows the curvatures for local Table 3.3: Hills data; curvatures using X + W perturbation. Model I contains DISTANCE and CLIMB. Model II contains DISTANCE and CLIMB-RATE Estimates in LD(w) or LD[O1](w) Model I Model II 2 Al = 30.47 A = 28.52 3 A1=21.15 A1=16.41 A2=16.49 A2=16.25 influence when fitting the regression model for winning time as a linear combination of DISTANCE and CLIMB. However, as was evident in Figure 1.5(c), the two regressors are linearly related. An alternative model can be fit by replacing climb in feet with climb in feet per distance travelled, CLIMB-RATE This regressor does not have the strong positive association with distance that the original coding of CLIMB has. This should reduce the collinearity in the model, thereby reducing sensitivity to perturbations in the X matrix. Using standardized regressors, the new model is fit. Column 2 of Table 3.3 shows the curvature values for this new model. Cma.x for directed influence on f has been substantially reduced, indicating that the MLEs for the second model are indeed less sensitive to perturbations in the X matrix. Figures 3.9(a), (b) and (c) provide the profile plots of the absolute value of the directional vectors that accompany the curvatures in Table 3.3, Column II. All three plots tend to indicate DISTANCE values as being more influential than CLIMB-RATE values, and all three plots show substantial change from those in Figure 3.8. The plot for &2 now indicates the DISTANCE value for observations 7 and 18 as influential, rather than just 18. No elements stand out in Vmax, while case 11's DISTANCE value is a very substantial contributor to v2. 1 0.75 0.5 0.25 S10. 20 .*.30 *'4-0 *5'- **69 *. W -0.25 -0.5 -0.75 -1 Figure 3.7: Hills data X + W pert.; vma for &2 Example 2 We now consider applying the X + W perturbation to the school data and examining directed influence on (3. The purpose of this example is to demonstrate alternative methods for displaying Vma,, when the number of regressors and/or observations is large. Figure 3.10 is the plot of Vma. for directed influence on /3. Elements 1-20 correspond to SALARY, elements 21-40 correspond to WCOLR, elements 41-60 correspond to SES, elements 61-80 correspond to TSCORE, and elements 81-100 correspond to MOMED. The two regressors that have little association with other explanatory variables, SALARY and TSCORE, have small elements. In contrast, some of the elements for the other regressors are more substantial. This plot does a fair job of highlighting regressors, but a poor job of highlighting observations. Figure 3.11 is the profile plot of Ivmaxl. It is easy to see that cases 3 and 18 have multiple regressor values that are influential. In addition, one of the values for case 2 is influential. However, is to difficult to keep track of which line represents which column. Another option is to plot "profiles" of each row, as in Figure 3.12. Here, some of the elements of observations 2, 3, and 18 are highlighted. 5 10 15 20 25 30 35 5 10 15 20 25 30 35 (a) IVmaxl for a2 5 10 15 20 25 30 35 (b) IVm.. for 0 5 10 15 20 25 30 35 (c) Iv21 for f Figure 3.8: Hills data; X + W pert.; column profiles of directional vectors. DIS- TANCE values are connected by solid line. CLIMB values are connected by a dashed line. 5 10 15 20 25 30 35 (a) IV|max for &2 5 10 15 20 25 30 35 (b) IVmaxl for 3 5 5 5 10 15 20 25 30 35 5 5 ( 1 (C) IV21 for Figure 3.9: Hills data; Model II; X+ W pert.; column profiles for directional vectors. DISTANCE values are connected by solid line. CLIMB-RATE values are connected by a dashed line. * a . "" 2Q 40. * * * - ~ .- -a *.6Q 80 * a Figure 3.10: School data; X + W pert.; Vmax for directed influence on f3 0.4 0.3 0'.2 i~~ ~ ,J/ ' 5 10 15 20 Figure 3.11: School data; X+W pert.; Column profiles of VmaxI for directed influence on /. Thick solid line- SALARY, thin solid line- WCOLR, dot-dashed line- SES, small-dashed line- TSCORE, large-dashed line- MOMED. 0.4[ 0.2 -0.2 -0.4 * 160 2 3 4 5 Figure 3.12: School data; X + W pert.; row profiles of IVmaxl for directed influence on /3 with large elements labeled. A bubble plot of I Wmax I provides the best graphical display of the information in Vma, (Figure 3.13). Each element is represented by a point plotted according to both the corresponding regressor (x-axis) and case (y-axis). Around each point is a bubble with radius equal to the absolute value of the element, and so large bubbles indicate large contributors to the direction of maximum curvature. Again, we see that the elements corresponding to x18is,2, x18is,5, x3,2, x3,5, and x2,3 have the largest influence, as evidenced by having the largest bubbles. 86 Row 20 0 0 0 0 a C. 15 0. 10 * *0*~ * 10 . 0 0 0 0 * S1 2 3 4 5 Column Figure 3.13: School data; X + W pert.; bubble plot of IWmax for 3 CHAPTER 4 LOCAL ASSESSMENT FOR OPTIMAL ESTIMATING FUNCTIONS It is possible to examine the local influence of perturbations on measures other than the likelihood displacement, LD(w), and the profile likelihood displacement, LD[G1](w). Several authors have applied local influence techniques to specific measures of influence. For instance, Lawrance (1988) and Wu and Luo (1993c) looked at influence on the Box-Cox transformation parameter. Also, influence has been examined on the residual sums of squares in regression (Wu and Luo 1993b), deviance residuals and confidence regions in GLMs (Thomas and Cook 1990; Thomas 1990), and goodness-of-link test statistics in GLMs (Lee and Zhao 1997). However, considerably less attention has been given to assessing local influence when estimation is not performed by ML. Some work has been done on extentions to Bayesian analysis (McCulloch 1989; Lavine 1992; Pan et al. 1996), and Lp norm estimation (Schall and Gonin 1991). In this chapter, local influence techniques are extended in two ways. First, we consider the influence of perturbations on a general class of influence measures. Second, we address parameter estimates that are found via estimating equations. The intent is to unify nearly all previous work on this methodology and provide a conceptual basis and computational tools for future applications. The first three sections outline the inferential platform, a general class of influence measures, and assumptions about the perturbations. Next, techniques for local assessment of perturbations are discussed. In Section 4.4, we give some algebraic results for the two main diagnostic quantities: the gradient vector and the acceleration matrix. Section 4.5 contains computational formulas, while Sections 4.6-4.8 contain applications. Several measures, including LD(w) and LD['] (w), are considered in section 4.6 using ML. Section 4.7 addresses influence on the generalized variance with an emphasis on ML. Finally, the last section considers quasi-likelihood. 4.1 Inferential Platform This section outlines assumptions about estimation, the nature of perturbations, and how the effects of perturbation are measured. 4.1.1 Estimating Functions Suppose that a statistical analysis is performed to estimate p parameters, 0. Further, suppose that point estimates, 0, are obtained as solutions to a set of p unbiased estimating equations, h(O; y) = 0, (4.1) where for each estimating function, hi(O; y), Eo(hi(O;y)) =0 i= 1,...p. Sometimes these estimating equations arise from minimizing or maximizing some criterion with respect to 0. Examples include least-squares normal equations, score equations and equations resulting from minimizing a Bayes risk. Quasi-likelihood (QL) equations (Wedderburn 1974) were originally motivated by maximizing a quasi-likelihood function, but the existence of the objective function is not necessary. Generalized estimating equations (GEEs) (Liang and Zeger 1986), like QL equations, merely require assumptions about first and second moments. The theory of estimating equations (EEs) dates back to early work by V. P. Godambe (Godambe 1960; Godambe 1976). There has been renewed interest in the subject due to new estimation techniques for semi-parametric models and models for clustered non-normal responses (Zeger and Liang 1986; Prentice 1988; McCullagh and Nelder 1989; Desmond 1997a). A considerable amount of work has centered on obtaining optimal estimating functions (OEFs) (Godambe 1976; Crowder 1986; Crowder 1987; Godambe and Heyde 1987; Firth 1987; Godambe and Thompson 1989; Godambe 1991). To this end, the efficiency matrix of a set of p estimating functions is defined as Eff(h) = (E [ ] )-1V[h](E h (4.2) Functions that minimize this non-negative definate matrix are considered optimal. Roughly speaking, this index of efficiency is small when the variance of the estimating functions is small and the average gradient is large. Provided that the estimating equations arise from maximizing or minimizing some function, we may interpret that large gradients indicate the function is not flat around its extremum. Assuming that a fully parameteric form for the data is assumed, the score functions are the unique OEFs up to a scalar multiple (Godambe 1960; Bhapkar 1972). However, with fewer assumptions about the data, it is generally not possible to find a unique set of optimal estimating functions. Instead, optimality is bestowed upon estimating functions within a certain class. One such class of estimating functions are those that are linear functions of the data. Under mild regularity conditions, the asymptotic variance of 0 is in fact given by (4.2) (Morton 1981; McCullagh 1983; Crowder 1986). Thus, minimizing Eff(h) is equivalent to minimizing the asymptotic variance of the parameter estimates. Note that the first interpretation of the efficiency matrix is appealing in that it is a finite sample index, but somewhat awkward in that it is based on a property of the estimating functions. The second interpretation is appealing in that it is based on a property of the estimates, but limited in that it requires asymptotics. For the purposes of applying local influence diagnostics we assume only that the estimating functions have been standardized. That is, given a set of unbiased estimating functions, h*(0; y), we utilize the standardized estimating functions given by h(O;y) =-E [-I-O (V [h*])-lh'. (4.3) This standardization has three implications. First, V(h(0; y))-1 is equivalent to (4.2) and hence equivalent to the asymptotic variance of the parameter estimates. In addition, the standardized estimating functions are optimal in the following sense. Given a set of unbiased estimating functions, {h(O; y)}, the OEFs amongst the class of linear combinations of the h* are given by (4.3). Hence, the standardized estimating functions are an optimal rescaling of the original estimating functions. Note that the negative sign is present in (4.3) so that the score equations, rather than the negative of the score equation, are considered standardized. Finally, standardized estimating functions are "information unbiased" (Lindsay 1982), meaning that the following equality holds: -E [Oh = E[hh'] = V[h]. (4.4) Hence, V[0] = -E[Oh/0O']. This fact is used later in the chapter to derive influence diagnostics for confidence ellipsoids. 4.1.2 Perturbations Perturbations are again introduced as a vector of q real numbers, Wo. These perturbations may affect the data, the model and its assumptions, or perhaps even just the estimation technique itself. Examples of data perturbations are minor changes or rounding errors in the response and the explanatory variables. An example of model perturbation is the variance perturbation scheme used in Chapters 1 and 2. Finally, perturbing the contribution of each observation to the estimating functions by case-weights is an example of perturbing the estimation technique. This perturbation does not affect the distributional assumptions of the data, but rather how each observation is utilized for inference. Another perturbation to the inference method would be to allow the value of the hyper-parameters in a Bayesian analysis to vary. Though some might argue that the hyper-parameters are part of the model, their values are often chosen for computational convenience. For the purposes of this chapter, we assume that the perturbation scheme results in a set of p perturbed estimating equations, h,(0; y) = 0, (4.5) which have solutions 6,. It is also assumed that there exists a null perturbation, wo, whose estimating equations are identical to the original ones. Finally, it is assumed that there are no constraints on w that would limit their values in a neighborhood of w0. For example, constraining the space of variance perturbations to positive values is allowed. However, constraining the perturbations to sum to one is not allowed. These kinds of constraints could be accommodated, but are not considered here. 4.1.3 Influence Measures The final item to specify is a quantity whose sensitivity to perturbation is of interest. This influence measure serves as a barometer for how perturbation effects inference. The measure can depend on w directly and indirectly via some r x 1 vector of functions of the perturbed parameter estimates, -y(O0,). Some influence measures, m(w, ,(O)), may be appropriate regardless of the method of estimation and the perturbation, while others may be more application-specific. Examples of rather general objective measures are the following: ml = x'16 (4.6) M= (4.7) m3 (Y (48) where Ew(yi) = p. None of these three measures depend on w directly. The first is a linear combination of the perturbed parameters. Here, 7(0O,) = z'O,, is a scalar. The second measure is the ratio of the ith unperturbed parameter estimate to the ith perturbed estimate. Again, y(9w) = 6L is a scalar. The third measure resembles a Pearson goodness-of-fit statistic based on the perturbed estimates. Since it is not yet specified how the means, fA,, depend on the estimates, we simply write y(0b,) = w. The following measures are more application-specific, in that they depend on how inference is performed and perhaps even upon the perturbation itself: m4 = LD(w) = 2[L(0; y) L(0,; y)] (4.9) m5 = LD'\(w) = 2[L(01,62; y) L(01,, g(10,);y)] (4.10) m6 = LD*(w) = 2[L,(6,; y) L0,(; y)] (4.11) f(y I 0) ] m7 = E [1og(f/(Y I 00) (4.12) m8 = log (h(O ;Y) (4.13) The first of these measures is the likelihood displacement, for which "Y(0,) = 0b. The second is the profile likelihood displacement, which has -Y(6O)' = ( g(06i)'), where g(01.) maximizes the profile likelihood of 01, with respect to 02. The third measure is a likelihood displacement with a "moving frame" because the base likelihood of the measure is the perturbed one. Obviously, these three measures are intended for use when estimation is performed by maximum likelihood. The influence measure m7 is a Kullback-Leibler divergence between the densities f(y I 0) and f(y | 0I). The quantity is non-negative, achieving its minimum value of zero when the two densities are equivalent. Here, Y'(0w) = 0Ow. The last measure addresses sensitivity of asymptotic standard errors. Recall that the determinant of a matrix equals the product of its eigenvalues, and when the matrix is a variance-covariance matrix for a vector of estimates 0, the squared |

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Section 4.7 addresses influence on the generalized variance with an emphasis on ML. Finally, the last section considers quasi-likelihood. 4.1 Inferential Platform This section outlines assumptions about estimation, the nature of perturbations, and how the effects of perturbation are measured. 4.1.1 Estimating Functions Suppose that a statistical analysis is performed to estimate p parameters, 0. Further, suppose that point estimates, 0, are obtained as solutions to a set of p unbiased estimating equations, h(0,y) = 0, (4.1) where for each estimating function, /(0; y), E9(hi{0;y)) = 0 i = l,...p. Sometimes these estimating equations arise from minimizing or maximizing some criterion with respect to 6. Examples include least-squares normal equations, score equations and equations resulting from minimizing a Bayes risk. Quasi-likelihood (QL) equations (Wedderburn 1974) were originally motivated by maximizing a quasi-likelihood function, but the existence of the objective function is not necessary. Generalized estimating equations (GEEs) (Liang and Zeger 1986), like QL equations, merely require assumptions about first and second moments. The theory of estimating equations (EEs) dates back to early work by V. P. Godambe (Godambe 1960; Godambe 1976). There has been renewed interest in the subject due to new estimation techniques for semi-parametric models and models for clustered non-normal responses (Zeger and Liang 1986; Prentice 1988; McCullagh and Nelder 1989; Desmond 1997a). A considerable amount of work has centered 126 4.8.1 Quasi-likelihood Consider a model in which a set of n independent observations y depends on a set of p parameters 0 through their means n(0). Although full distributional assumptions are not made, we also assume that the variances of y depend on their means: V[y] = cr2V(n). A class of estimating functions that we may consider for estimating 6 are given by h'(e-,y) = W(0)(y-n(e))/c2, (4.71) where the p x n weight matrix W (0) can be a function of the unknown parameters. Standardizing (4.71) as in (4.3) yields (Wedderburn 1974; Godambe and Heyde 1987): h{0] y) = %-V(v)~1(y V{0))/o2- (4-72) These optimal estimating functions are called the quasi-score functions because the solutions to h(0,y) = 0 maximize the following quasi-likelihood function, provided the integral exists: Q = [^T)dt (473) 4.8.2 Local Assessment Some preliminary results for local assessment of quasi-likelihood models for independent data are presented in this section. Assumptions about the models form are presented, some perturbation schemes are proposed, and a case-weight perturbation is applied to an example data set. 4.8.2.1 Assumptions It is assumed that observations are independent, and that the variance is related to the mean by the function V(pi). In addition, it is assumed that the mean 200 Thomas, W. (1991), Influence Diagnostics for the Cross-validated Smoothing Parameter in Spline Smoothing, Journal of the American Statistical Association, 86, 693-698. Thomas, W. and Cook, R. D. (1989), Assessing Influence on Regression Coefficients in Generalized Linear Models, Biometrika, 76, 741-749. Thomas, W. and Cook, R. D. (1990), Assessing Influence on Predictions From Generalized Linear Models, Technometrics, 32, 59-65. Tsai, C.-L. and Wu, X. (1992), Assessing Local Influence in Linear Regression Models With First-order Autoregressive Or Heteroscedastic Error Structure, Statistics & Probability Letters, 14, 247-252. Vos, P. W. (1991), A Geometric Approach to Detecting Influential Cases, The Annals of Statistics, 19, 1570-1581. Wang, S.-J. and Lee, S.-Y. (1996), Sensitivity Analysis of Structural Equation Models With Equality Functional Constraints, Computational Statistics and Data Analysis, 23, 239-256. Wang, X., Ren, S., and Shi, L. (1996), Local Influence in Discriminant Analysis, Journal of Systems Science and Mathematical Science, 8, 27-36. Wedderburn, R. W. M. (1974), Quasi-likelihood Functions, Generalized Linear Models, and the Gauss-Newton Method, Biometrika, 61, 439-447. Weiss, R. (1996), An Approach to Bayesian Sensitivity Analysis, Journal of the Royal Statistical Society, Series B, 58, 739-750. Weissfeld, L. A. (1990), Influence Diagnostics for the Proportional Hazards Model, Statistics & Probability Letters, 10, 411-417. Weissfeld, L. A. and Schneider, H. (1990), Influence Diagnostics for the Weibull Model Fit to Censored Data, Statistics & Probability Letters, 9, 67-73. Wu, X. and Luo, Z. (1993a), Residual Sum of Squares and Multiple Potential, Diagnostics By a Second Order Local Approach, Statistics & Probability Letters, 16, 289-296. Wu, X. and Luo, Z. (1993b), Residual Sum of Squares and Multiple Potential, Diagnostics By a Second Order Local Approach, Statistics & Probability Letters, 16, 289-296. Wu, X. and Luo, Z. (1993c), Second-Order Approach to Local Influence, Journal of the Royal Statistical Society, Series B, 55, 929-936. Wu, X. and Wan, F. (1994), A Perturbation Scheme for Nonlinear Models, Statistics & Probability Letters, 20, 197-202. 109 4.5.1.1 Remarks The formula for P can be combined with the results of Theorem 4.2 to further revise the acceleration matrix: d2m(u),'y(0UJ)) M = dujduj1 I J'K' I J'K' I KJ I KJ + [*;<8> J'] GJ+[s\KIq]P + [; j'} gj + [s'^K Iq\ 1 J -ft [ J, J' ] - J I J'K' + I KJ + [; j'] gj . 1 I s\Kh [ lq J' } J (4.44) Also, we may write p [s'^K Iq]P = Y^(s'iKi)Pi i=l P = Â£(-*>) 3=1 =1 p p j=1 =1 /9 J' Iq J' *3 *9 J Â£ W-*; 3-1 J, J' (4.45) 136 E(ti|y) = E(w) is Z/3, and its mean squared error of prediction can be shown to be a2(Z'(X'X)~1Z + /m). Also, Z(3 is normally distributed. These facts can be used to create a 100(1 a)% prediction interval for a single u with covariate vector z\ z'P Za/2a2(z'(X'X)~lz -f 1). (5.3) In the event that a2 is unknown, the prediction interval given in (5.3) is still valid but cannot be calculated. One option is to simply replace <72 by s2 to produce an approximate prediction interval. A better alternative is to make use of the following pivotal quantity: u z'Â¡3 '-nki y/s2(z'(X'X)~1Z + 1) where k is the number of regressors. An exact 100(1 a)% prediction interval is then (5.4) z'0 /2^sHz'(X'X)-'z + 1). (5.5) This simple situation highlights the fact that unknown parameters can complicate the prediction problem. While pivotal quantities may be available in simple situations, accomodating unknown parameters may be more cumbersome in complicated models. Therefore, attempts have been made to unify predictive inference via predictive likelihood. 5.2 Predictive Likelihood Predictive likelihood is a likelihood-based framework for handling the problem of prediction. The goal is to predict u via a point or interval predictor in the presence of unknown parameters 6. If u is treated as a vector of parameters, then the joint likelihood of u and d is L'(u, 8, y) = f(y, u \ 8) = f,(u | 8, y)f2(y | 8). (5.6) 38 2.6.2.1 1-to-l coordinate changes in ft Following Loynes (1986) and Beckman et al. (1987), let us consider 1-to-l coordinate changes in the perturbation space 2. That is, a new perturbation scheme is constructed with elements = k(ui) for i 1,... q, for a smooth function k. The new schemes perturbation, uj*, will not yield the same curvature as a;. Specifically, Cu* = (Â§Â£:)w_ Cu, where uio denotes the null perturbation value for w*. 2 2 As an example, consider perturbing the variance of yi to be % rather than 7-. Here, k(uji) uj?, resulting in new curvatures that are (Â§^U;=i)2 = (2u;)2.=1 = 4 times the original ones. Hence, despite the fact that the two versions of 7 can produce the same set of perturbed models, the curvature in identical directions will change. This implies that a universal benchmark for curvatures is not possible. Indeed, given the arbitrariness of scaling the perturbation, it should come as no surprise that curvatures cannot be compared across perturbation schemes. However, the direction of maximum curvature is unchanged under 1-to-l coordinate changes in 2. 2.6.2.2 Interpreting curvatures In this section, a simple way to assess curvatures from the same perturbation scheme is given. First, y(7 is a Taylor series approximation to LD(u)0 + av) (Lawrance 1991; Escobar and Meeker 1992). Also, assuming that n is large, an asymptotic 100(1 a)% likelihood-based confidence region for 0 is given by values of 9 such that 2(L(0;y) L(Q-y)) < xl,i-a- Thus, if twice the curvature in direction v is larger than xl,i-Q, then a perturbation at a distance of one from u?o in direction v moves the parameter estimates to the edge of the asymptotic confidence region. 79 Example 1 Consider the Scottish hills data set described in Chapter 1. The purpose of this example is twofold- to demonstrate how the X-matrix results can be displayed graphically, and to demonstrate how collinearity affects the curvature values when examining influence on $. Figure 3.7 gives the direction of maximum curvature for directed influence on a2. The first 35 elements correspond to perturbations of the DISTANCE regressor values, while elements 36-70 correspond to perturbations of the CLIMB regressor values. The value for race 18s DISTANCE is the largest contributer to this influential perturbation. Elements 7 (DISTANCE for race 7) and 53 (CLIMB for race 18) are also large The information in Figure 3.7 can be difficult to discern because the perturbation vector enters into the likelihood in the form of a matrix, W. Another way to display umax is to first take its absolute value and then to consider the elements corresponding to each column of W. Each portion can then be plotted against indices 1,..., n and shown together (Figure 3.8(a)). This graphical display provides a profile plot of each column of Ws contribution to the direction of maximum curvature. Here we see again that elements corresponding to observations 18 and 7 as being the most influential. There is also some evidence that measurement error in the DISTANCE values (solid line) are more influential than error in the CLIMB values (dashed line). Figures 3.8(b) and 3.8(b) provide profile plots of the absolute value of the .. r attention to observations 7, 11, and 18, while no observations appear noteworthy in the second plot. At this point we turn our attention to the curvature values that accompany these directional vectors (Table 3.3). The first column shows the curvatures for local 28 2.4.1 Building Blocks The perturbed likelihood is 77 Lu(0; V) oc -- loga2 ~ X/3 + u)\y X0 + u) = log o'2 ^(y + w)'(y + w) Tt 1 = --loga2 2^2(y'y + 2w'y + w'w), where y = y X(3. The next step is construction of A. Partial derivatives with respect to u are as follows: d Lu(9\y) du ~(y-xÂ¡8 + ). Second partial derivatives are then dud Â¡3' d2Lu{0\y) duda2 = -tX = -7(y X^ + u;). Evaluating at the original model yields &Lu{9-,y) dud/3' d2Lu(0;y) = X G=0,u=uo 1 = T-rV. G=6,u=u0 duda2 From these quantities, the n x p matrix A' can be constructed: 1 A' = X hr Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy LOCAL ASSESSMENT OF PERTURBATIONS By Glen Lawson Hartless August 2000 Chairman: James G. Booth Major Department: Statistics Statistical models are useful and convenient tools for describing data. However, models can be adversely effected by mistaken information, poor assumptions, and small subsets of influential data. Seemingly inconsequential changes to the observed data, the postulated model, or the inference technique may greatly affect the scientific conclusions made from model-fitting. This dissertation considers local assessment of perturbations in parametric and semi-parametric models. The effects of perturbation are assessed using a Taylor-series approximation in the vicinity of the fitted model, leading to graphical diagnostics that allow the analyst to identify sensitive parameter estimates and sources of undue influence. First, computational formulas are given to assess the influence of perturbations on linear combinations of the parameter estimates for maximum likelihood. A diagnostic plot that identifies sensitive fitted values and self-predicting observations in linear regression is introduced. Second, the eigensystem of the acceleration matrix for perturbing each element of the design matrix in linear regression is derived. vii 67 From these quantities we can construct the np x p matrix A', as given by Cook (1986): Ikr-j3X (3.21) where <8> denotes the Kronecker product (Searle 1982). The next step is to construct the acceleration matrix. Here we use the fact that (A (g> B)(C D) AC Noting that A = A\ = \ A, the acceleration matrix can be written as F = (/* = ^ (ijk <2) r P (3.22) 3.4.2 Directed Influence on d2 The acceleration matrix for the profile likelihood displacement for 4 nu4 /3P <8> rr'. (3.23) This matrix has a single non-zero eigenvalue, -^-||/3||2, with corresponding eigenvector oc (3 r. This solution is confirmed below: (F1* j AI)v oc 43p'rr'-^\\p\\2I ($r) no o no* no4 PP' rr' -\\P\\2\\r\\2I (Â¡3 \\P\\2P IM|2r ||/9||2||r||2(j9 r) = 0. (3.24) 77 Proof of Theorem 3.2 We begin by obtaining update formulas for A and L 1 as per Section 2.7: = = 0X)
= r Â¡3 Xip)
ipi = the ith eigenvector of X'X.
model. The new regressors, Zx = Xtpx, ...,Zk = X(pk, are known as the principal
weights are the first residual (ri) and the negative of the first observations value for
a: =
ki k2 ...
is a full rank pxp matrix with
, r-i/2
-Lj q X
k'ikj =0 i^j
k[ki= k[ki i = 2...p.
By construction,
1 = x'6
0 = tJ^L?/2K(I>
K =
do
IIM2
K'
1 l:1/2k.
,|fei||2 0 V d |